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Unformatted text preview: arXiv:hepth/0110055 v3 3 Jan 2002 Strings, Branes and
Extra Dimensions
Stefan F¨rste
o Physikalisches Institut, Universit¨t Bonn
a
Nussallee 12, D53115 Bonn, Germany Abstract This review is devoted to strings and branes. Firstly, perturbative string theory is
introduced. The appearance of various types of branes is discussed. These include
orbifold ﬁxed planes, Dbranes and orientifold planes. The connection to BPS vacua
of supergravity is presented afterwards. As applications, we outline the role of branes
in string dualities, ﬁeld theory dualities, the AdS/CFT correspondence and scenarios
where the string scale is at a TeV. Some issues of warped compactiﬁcations are also
addressed. These comprise corrections to gravitational interactions as well as the
cosmological constant problem. Contents
1 Introduction 1 2 Perturbative description of branes
2.1 The Fundamental String . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Worldsheet Actions . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1.1 The closed bosonic string . . . . . . . . . . . . . . . .
2.1.1.2 Worldsheet supersymmetry . . . . . . . . . . . . . . .
2.1.1.3 Spacetime supersymmetric string . . . . . . . . . . .
2.1.2 Quantization of the fundamental string . . . . . . . . . . . . .
2.1.2.1 The closed bosonic string . . . . . . . . . . . . . . . .
2.1.2.2 Type II strings . . . . . . . . . . . . . . . . . . . . . .
2.1.2.3 The heterotic string . . . . . . . . . . . . . . . . . . .
2.1.3 Strings in nontrivial backgrounds . . . . . . . . . . . . . . . .
2.1.4 Perturbative expansion and eﬀective actions . . . . . . . . . . .
2.1.5 Toroidal Compactiﬁcation and Tduality . . . . . . . . . . . . .
2.1.5.1 KaluzaKlein compactiﬁcation of a scalar ﬁeld . . . .
2.1.5.2 The bosonic string on a circle . . . . . . . . . . . . . .
2.1.5.3 Tduality in non trivial backgrounds . . . . . . . . . .
2.1.5.4 Tduality for superstrings . . . . . . . . . . . . . . . .
2.2 Orbifold ﬁxed planes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The bosonic string on an orbicircle . . . . . . . . . . . . . . . .
2.2.2 Type IIB on T 4/Z2 . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Comparison with type IIB on K 3 . . . . . . . . . . . . . . . . .
2.3 Dbranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . .
2.3.1.2 Quantization of the open string ending on a single Dbrane . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1.3 Number of ND directions and GSO projection . . . .
2.3.1.4 Multiple parallel Dbranes – Chan Paton factors . . .
2.3.2 Dbrane interactions . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Dbrane actions . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3.1 Open strings in nontrivial backgrounds . . . . . . . .
2.3.3.2 Toroidal compactiﬁcation and Tduality for open strings
2.3.3.3 RR ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . .
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91 CONTENTS ii 2.3.3.4 Noncommutative geometry . . . . . . . . . . . . .
2.4 Orientifold ﬁxed planes . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Unoriented closed strings . . . . . . . . . . . . . . . . . . .
2.4.2 Oplane interactions . . . . . . . . . . . . . . . . . . . . . .
2.4.2.1 Oplane/Oplane interaction, or the Klein bottle .
2.4.2.2 Dbrane/Oplane interaction, or the M¨bius strip
o
2.4.3 Compactifying the transverse dimensions . . . . . . . . . .
2.4.3.1 Type I/type I strings . . . . . . . . . . . . . . . .
2.4.3.2 Orbifold compactiﬁcation . . . . . . . . . . . . . .
3 NonPerturbative description of
3.1 Preliminaries . . . . . . . . . .
3.2 Universal Branes . . . . . . . .
3.2.1 The fundamental string
3.2.2 The NS ﬁve brane . . .
3.3 Type II branes . . . . . . . . . branes
.....
.....
.....
.....
..... 4 Applications
4.1 String dualities . . . . . . . . . . . . . .
4.2 Dualities in Field Theory . . . . . . . .
4.3 AdS/CFT correspondence . . . . . . . .
4.3.1 The conjecture . . . . . . . . . .
4.3.2 Wilson loop computation . . . .
4.3.2.1 Classical approximation
4.3.2.2 Stringy corrections . . .
4.4 Strings at a TeV . . . . . . . . . . . . .
4.4.1 Corrections to Newton’s law . . . .
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165 5 Brane world setups
5.1 The Randall Sundrum models . . . . . . . . . . . .
5.1.1 The RS1 model with two branes . . . . . .
5.1.1.1 A proposal for radion stabilization
5.1.2 The RS2 model with one brane . . . . . . .
5.1.2.1 Corrections to Newton’s law . . .
5.1.2.2 ... and the holographic principle .
5.1.2.3 The RS2 model with two branes .
5.2 Inclusion of a bulk scalar . . . . . . . . . . . . . .
5.2.1 A solution generating technique . . . . . . .
5.2.2 Consistency conditions . . . . . . . . . . . .
5.2.3 The cosmological constant problem . . . . .
5.2.3.1 An example . . . . . . . . . . . . .
5.2.3.2 A no go theorem . . . . . . . . . . .
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. CONTENTS
6 Bibliography and further reading
6.1 Chapter 2 . . . . . . . . . . . . .
6.1.1 Books . . . . . . . . . . .
6.1.2 Review articles . . . . . .
6.1.3 Research papers . . . . .
6.2 Chapter 3 . . . . . . . . . . . . .
6.2.1 Review articles . . . . . .
6.2.2 Research Papers . . . . .
6.3 Chapter 4 . . . . . . . . . . . . .
6.3.1 Review articles . . . . . .
6.3.2 Research papers . . . . .
6.4 Chapter 5 . . . . . . . . . . . . .
6.4.1 Review articles . . . . . .
6.4.2 Research papers . . . . . iii .
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. 197
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202 Chapter 1 Introduction
One of the most outstanding problems of theoretical physics is to unify our picture of
electroweak and strong interactions with gravitational interactions. We would like to
view the attraction of masses as appearing due to the exchange of particles (gravitons)
between the masses. In conventional perturbative quantum ﬁeld theory this is not
possible because the theory of gravity is not renormalizable. A promising candidate
providing a uniﬁed picture is string theory. In string theory, gravitons appear together
with the other particles as excitations of a string.
On the other hand, also from an observational point of view gravitational interactions show some essential diﬀerences to the other interactions . Masses always attract
each other, and the strength of the gravitational interaction is much weaker than the
electroweak and strong interactions. A way how this diﬀerence could enter a theory
is provided by the concept of “branes. The expression “brane” is derived from membrane and stands for extended objects on which interactions are localized. Assuming
that gravity is the only interaction which is not localized on a brane, the special features of gravity can be attributed to properties of the extra dimensions where only
gravity can propagate. (This can be either the size of the extra dimension or some
curvature.)
The brane picture is embedded in a natural way in string theory. Therefore, string
theory has the prospect to unify gravity with the strong and electroweak interactions
while, at the same time, explaining the diﬀerence between gravitational and the other
interactions.
This set of notes is organized as follows. In chapter 2, we brieﬂy introduce the
concept of strings and show that quantized closed strings yield the graviton as a string
excitation. We argue that the quantized string lives in a ten dimensional target space.
It is shown that an eﬀective ﬁeld theory description of strings is given by (higher dimensional supersymmetric extensions of) the Einstein Hilbert theory. The concept
1 1. Introduction 2 of compactifying extra dimensions is introduced and special stringy features are emphasized. Thereafter, we introduce the orbifold ﬁxed planes as higher dimensional
extended objects where closed string twisted sector excitations are localized. The
quantization of the open string will lead us to the concept of Dbranes, branes on
which open string excitations live. We compute the tensions and charges of Dbranes
and derive an eﬀective ﬁeld theory on the world volume of the Dbrane. Finally,
perturbative string theory contains orientifold planes as extended objects. These are
branes on which excitations of unoriented closed strings can live. Compactiﬁcations
containing orientifold planes and Dbranes are candidates for phenomenologically interesting models. We demonstrate the techniques of orientifold compactiﬁcations at a
simple example.
In chapter 3, we identify some of the extended objects of chapter 2 as stable
solutions of the eﬀective ﬁeld theory descriptions of string theory. These will be the
fundamental string and the Dbranes. In addition we will ﬁnd another extended object,
the NS ﬁve brane, which cannot be described in perturbative string theory.
Chapter 4 discusses some applications of the properties of branes derived in the
previous chapters. One of the problems of perturbative string theory is that the string
concept does not lead to a unique theory. However, it has been conjectured that all
the consistent string theories are perturbative descriptions of one underlying theory
called Mtheory. We discuss how branes ﬁt into this picture. We also present branes
as tools for illustrating duality relations among ﬁeld theories. Another application, we
are discussing is based on the twofold description of three dimensional Dbranes. The
perturbative description leads to an eﬀective conformal ﬁeld theory (CFT) whereas the
corresponding stable solution to supergravity contains an AdS space geometry. This
observation results in the AdS/CFT correspondence. We present in some detail, how
the AdS/CFT correspondence can be employed to compute Wilson loops in strongly
coupled gauge theories. An application which is of phenomenological interest is the
fact that Dbranes allow to construct models in which the string scale is of the order of
a TeV. If such models are realized in nature, they should be discovered experimentally
in the near future.
Chapter 5 is somewhat disconnected from the rest of these notes since it considers
brane models which are not directly constructed from strings. Postulating the existence
of branes on which certain interactions are localized, we present the construction of
models in which the space transverse to the brane is curved. We discuss how an
observer on a brane experiences gravitational interactions. We also make contact to the
AdS/CFT conjecture for a certain model. Also other questions of phenomenological
relevance are addressed. These are the hierarchy problem and the problem of the
cosmological constant. We show how these problems are modiﬁed in models containing 1. Introduction 3 branes.
Chapter 6 gives hints for further reading and provides the sources for the current
text.
Our intention is that this review should be self contained and be readable by people
who know some quantum ﬁeld theory and general relativity. We hope that some people
will enjoy reading one or the other section. Chapter 2 Perturbative description of
branes
2.1
2.1.1
2.1.1.1 The Fundamental String
Worldsheet Actions
The closed bosonic string Let us start with the simplest string – the bosonic string. The string moves along a
surface through space and time. This surface is called the worldsheet (in analogy to a
worldline of a point particle). For space and time in which the motion takes place we
will often use the term target space. Let d be the number of target space dimesnions.
The coordinates of the target space are X µ , and the worldsheet is a surface X µ (τ, σ ),
where τ and σ are the time and space like variables parameterizing the worldsheet.
String theory is deﬁned by the requirement that the classical motion of the string
should be such that its worldsheet has minimal area. Hence, we choose the action of
the string proportional to the worldsheet. The resulting action is called Nambu Goto
action. It reads
S=− 1
2πα d2 σ √
−g. (2.1.1.1) The integral is taken over the parameter space of σ and τ . (We will also use the
notation τ = σ 0, and σ = σ 1.) The determinant of the induced metric is called g . The
induced metric depends on the shape of the worldsheet and the shape of the target
space,
gαβ = Gµν (X ) ∂α X µ∂β X ν , 4 (2.1.1.2) 2. Worldsheet Actions 5 where µ, ν label target space coordinates, whereas α, β label worldsheet parameters.
Finally, we have introduced a constant α . It is the inverse of the string tension and
has the mass dimension −2. The choice of this constant sets the string scale. By construction, the action (2.1.1.1) is invariant under reparametrizations of the worldsheet.
Alternatively, we could have introduced an independent metric γαβ on the worldsheet. This enables us to write the action (2.1.1.1) in an equivalent form,
S=− 1
4πα √
d2σ −γγ αβ Gµν ∂α X µ∂β X ν . (2.1.1.3) For the target space metric we will mostly use the Minkowski metric ηµν in the present
chapter. Varying (2.1.1.3) with respect to γαβ yields the energy momentum tensor,
1
4πα δS
= ∂αX µ ∂β Xµ − γαβ γ δγ ∂δ X µ∂γ Xµ ,
Tαβ = − √
αβ
−γ δγ
2 (2.1.1.4) where the target space index µ is raised and lowered with Gµν = ηµν . Thus, the γαβ
equation of motion, Tαβ = 0, equates γαβ with the induced metric (2.1.1.2), and the
actions (2.1.1.1) and (2.1.1.3) are at least classically equivalent. If we had just used
covariance as a guiding principle we would have written down a more general expression
for (2.1.1.3). We will do so later. At the moment, (2.1.1.3) with Gµν = ηµν describes
a string propagating in the trivial background. Upon quantization of this theory we
will see that the string produces a spectrum of target space ﬁelds. Switching on non
trivial vacua for those target space ﬁelds will modify (2.1.1.3). But before quantizing
the theory, we would like to discuss the symmetries and introduce supersymmetric
versions of (2.1.1.3).
First of all, (2.1.1.3) respects the target space symmetries encoded in Gµν . In
our case Gµν = ηµν this is nothing but d dimensional Poincar´ invariance. From
e
the two dimensional point of view, this symmetry corresponds to ﬁeld redeﬁnitions
in (2.1.1.3). The action is also invariant under two dimensional coordinate changes
(reparametrizations). Further, it is Weyl invariant, i.e. it does not change under
γαβ → eϕ(τ,σ) γαβ . (2.1.1.5) It is this property which makes one dimensional objects special. The two dimensional
coordinate transformations together with the Weyl transformations are suﬃcient to
transform the worldsheet metric locally to the Minkowski metric,
γαβ = ηαβ . (2.1.1.6) It will prove useful to use instead of σ 0 , σ 1 the light cone coordinates,
σ − = τ − σ , and σ + = τ + σ. (2.1.1.7) 2. Worldsheet Actions 6 So, the gauged ﬁxed version1 of (2.1.1.3) is
S= 1
2πα dσ + dσ − ∂− X µ∂+ Xµ . (2.1.1.8) However, the reparametrization invariance is not completely ﬁxed. There is a residual
invariance under the conformal coordinate transformations,
σ+ → σ+ σ+
˜ , σ− → σ− σ− .
˜ (2.1.1.9) This invariance is connected to the fact that the trace of the energy momentum tensor
(2.1.1.4) vanishes identically, T+− = 02 . However, the other γαβ equations are not
identically satisﬁed and provide constraints, supplementing (2.1.1.8),
T++ = T−− = 0. (2.1.1.10) The equations of motion corresponding to (2.1.1.8) are3
∂+ ∂− X µ = 0 (2.1.1.11) Employing conformal invariance (2.1.1.9) we can choose τ to be an arbitrary solution
to the equation ∂+ ∂− τ = 0. (The combination of (2.1.1.9) and (2.1.1.7) gives
τ→ 1++
˜
σ σ + σ− σ−
˜
2 , (2.1.1.12) which is the general solution to (2.1.1.11)). Hence, without loss of generality we can
ﬁx
1
X + = √ X 0 + X 1 = x+ + p+ τ,
2 (2.1.1.13) where x+ and p+ denote the center of mass position and momentum of the string in
the + direction, respectively. The constraint equations (2.1.1.10) can now be used to
ﬁx
1
X− = √ X0 − X1
2 (2.1.1.14) as a function of X i (i = 2, . . . , d − 1) uniquely up to an integration constant corresponding to the center of mass position in the minus direction. Thus we are left with
1 Gauge ﬁxing means imposing (2.1.1.6).
The corresponding symmetry is called conformal symmetry. It means that the action is invariant under conformal coordinate transformations while keeping the worldsheet metric ﬁxed. In two
dimensions this is equivalent to Weyl invariance.
3
For the time being we will focus on closed strings. That means that we impose periodic boundary
conditions and hence there are no boundary terms when varying the action. We will discuss open
strings when turning to the perturbative description of Dbranes in section 2.3.
2 2. Worldsheet Actions 7 d − 2 physical degrees of freedom X i. Their equations of motion are (2.1.1.11) without
any further constraints. By employing the symmetries of (2.1.1.3) we managed to
reduce the system to d − 2 free ﬁelds (satisfying (2.1.1.11)). Since these symmetries
may suﬀer from quantum anomalies we will have to be careful when quantizing the
theory in section 2.1.2.
2.1.1.2 Worldsheet supersymmetry In this section we are going to modify the previously discussed bosonic string by
enhancing its two dimensional symmetries. We will start from the gauge ﬁxed action (2.1.1.8) which had as residual symmetries two dimensional Poincar´ invariance
e
4 A natural extension of Poincar´
and conformal coordinate transformations (2.1.1.9).
e
invariance is supersymmetry. Therefore, we will study theories which are supersymmetric from the two dimensional point of view. In order to construct a supersymmetric extension of (2.1.1.8) one should ﬁrst specify the symmetry group and then use
Noether’s method to build an invariant action. We will be brief and just present the
result,
S= 1
2πα iµ
iµ
dσ +dσ − ∂− X µ∂+ Xµ + ψ+ ∂− ψ+µ + ψ− ∂+ ψ−µ ,
2
2 (2.1.1.15) where ψ± are two dimensional MajoranaWeyl spinors. To see this, we ﬁrst note that
ψ−
ψ+ 1
iψ+ ∂− ψ+ + iψ− ∂+ ψ− = − (ψ+ , −ψ−) ρ+ ∂+ + ρ− ∂−
2 , (2.1.1.16) where
ρ± = ρ 0 ± ρ 1 , (2.1.1.17) with
ρ0 = 0 −i
i 0 and ρ1 = 0i
i0 . (2.1.1.18) It is easy to check that the above matrices form a two dimensional Cliﬀord algebra,
ρα, ρβ = −2η αβ .
Also, note that i (ψ+ , −ψ− ) is the Dirac conjugate of
Majorana spinor ψ−
ψ+ (2.1.1.19)
ψ−
ψ+ for real ψ± , i.e. of the . In addition to two dimensional Poincar´ invariance and
e 4
Alternatively, we could start from the action (2.1.1.3). This we would modify such that it becomes
locally supersymmetric. Finally, we would ﬁx symmetries in the locally supersymmetric action. 2. Worldsheet Actions 8 invariance under conformal coordinate transformations (2.1.1.9)5 the action (2.1.1.15)
is invariant under worldsheet supersymmetry,
δX µ = ¯ψ µ = i µ
+ ψ− µ
− ψ+ , −i δψ µ = −iρα∂α X µ . (2.1.1.20)
(2.1.1.21) In components (2.1.1.21) gives rise to the two equations
µ
δψ− = −2
µ
δψ+ =2 + ∂− X − ∂+ X µ µ , (2.1.1.22) . (2.1.1.23) When checking the invariance of (2.1.1.15) under (2.1.1.20), (2.1.1.22), (2.1.1.23) one
should take into account that spinor components are anticommuting, e.g. + ψ− =
−ψ− + . Since the supersymmetry parameters ± form a non chiral Majorana spinor, the above symmetry is called (1, 1) supersymmetry. (In the end of this section we will
also discuss the chiral (1, 0) supersymmetry.) To summarize, the action (2.1.1.15) has
the following two dimensional global symmetries: Poincar´ invariance and supersyme
metry. The corresponding Noether currents are the energy momentum tensor,
iµ
T++ = ∂+ X µ∂+ Xµ + ψ+ ∂+ ψ+µ ,
2
iµ
T−− = ∂− X µ∂− Xµ + ψ− ∂− ψ−µ ,
2 (2.1.1.24)
(2.1.1.25) and the supercurrent
µ
J+ = ψ+ ∂+ Xµ , (2.1.1.26) µ J− = ψ− ∂− Xµ . (2.1.1.27) The vanishing of the trace of the energy momentum tensor T+− ≡ 0 is again a consequence of the invariance under the (local) conformal coordinate transformations
3
(2.1.1.9). The supercurrent is a spin– 2 object and naively one would expect to get
four independent components. That there are only two nonvanishing components is
a consequence of the fact that the supersymmetries (2.1.1.20), (2.1.1.22), (2.1.1.23)
leave the action invariant also when we allow instead of constant ± for
− = − σ+ and + = + σ− , (2.1.1.28) i.e. they are “partially” local symmetries. Once again, the vanishing of the energy
momentum tensor is an additional constraint on the system. We did not derive this
5 Under the transformation (2.1.1.9) the spinor components transform as ψ± → σ±
˜ −1
2 ψ± . 2. Worldsheet Actions 9 explicitly here. But it can be easily inferred as follows. In two dimensions the Einstein
tensor vanishes identically. Thus, if we were to couple to two dimensional (Einstein)
gravity, the constraint Tαβ = 0 would correspond to the Einstein equation. Similarly, the supercurrents (2.1.1.26), (2.1.1.27) are constrained to vanish. (If the theory
was coupled to two dimensional supergravity, this would correspond to the gravitino
equations of motion.)
As in the bosonic case we can employ the symmetry (2.1.1.9) to ﬁx
X + = x+ + p+ τ.
The local supersymmetry transformation (2.1.1.21) with
used to gauge
ψ−
ψ+ (2.1.1.29)
given by (2.1.1.28) can be µ=+ = 0. (2.1.1.30) (We have written here the target space (light cone) index as µ = + in order to avoid
confusion with the worldsheet spinor indices.) Note, that the gauge ﬁxing condition (2.1.1.30) is compatible with (2.1.1.29) and the supersymmetry transformations
(2.1.1.20), (2.1.1.21), as (2.1.1.30) implies the supersymmetry transformation
δX + = 0. (2.1.1.31) The constraints (2.1.1.24), (2.1.1.25), (2.1.1.26), (2.1.1.27) can be solved for X − , and
µ
ψα=− (here, α denotes the worldsheet spinor index). Therefore, after ﬁxing the local
symmetries completely we are left with d − 2 free bosons and d − 2 free fermions (from
a two dimensional point of view).
We should note that in the closed string case (periodic boundary conditions in
bosonic directions) we have two choices for boundary conditions on the worldsheet
fermions. Boundary terms appearing in the variation of the action vanish for either
periodic or anti periodic boundary conditions on worldsheet fermions. (Later, we will
call the solutions with antiperiodic fermions Neveu Schwarz (NS) sector and the ones
with periodic boundary conditions Ramond (R) sector.
Going back to (2.1.1.15), we note that alternatively we could have written down
µ
a (1, 0) supersymmetric action by setting the left handed fermions ψ+ = 0. The
supersymmetries are now given by (2.1.1.20) and (2.1.1.22), only. The parameter −
does not occur anymore, and hence we have reduced the number of supersymmetries by
one half. More generally one can add left handed fermions λA which do not transform
+
under supersymmetries. Therefore, they do not need to be in the same representation
of the target space Lorentz group as the X µ (therefore the index A instead of µ). 2. Worldsheet Actions 10 Summarizing we obtain the following (1, 0) supersymmetric action
1
S=
2πα + dσ dσ − i
iµ
∂− X ∂+ Xµ + ψ− ∂+ ψ−µ +
2
2 N µ A=1 λA ∂− λ+A
+ . (2.1.1.32) this will turn out to be the worldsheet action of the heterotic string. The energy moµ
mentum tensor is as given in (2.1.1.24), (2.1.1.25) with λA replacing ψ+ in (2.1.1.25).
+
There is only one conserved supercurrent (2.1.1.26).
Finally, we should remark that there are also extended versions of two dimensional supersymmetry (see for example [456]). We will not be dealing with those in this
review.
2.1.1.3 Spacetime supersymmetric string In the above we have extended the bosonic string (2.1.1.3) to a superstring from the two
dimensional perspective. We called this worldsheet supersymmetry. Another direction
would be to extend (2.1.1.3) such that the target space Poincar´ invariance is enhanced
e
to target space supersymmetry. This concept leads to the Green Schwarz string. Space
time supersymmetry means that the bosonic coordinates X µ get fermionic partners θA
(where A labels the number of supersymmetries N ) such that the targetspace becomes
a superspace. In addition to Lorentz symmetry, the supersymmetric extension mixes
fermionic and bosonic coordinates,
δθA = A ,
¯
δ θ = ¯A ,
δX µ (2.1.1.33)
(2.1.1.34)
µA = i¯Γ θ , (2.1.1.35) where the global transformation parameter A is a target space spinor and Γµ denotes
a target space Dirac matrix. In order to construct a string action respecting the
symmetries (2.1.1.33) – (2.1.1.35) one tries to replace ∂αX µ by the supersymmetric
combination
¯
Πµ = ∂α X µ − iθA Γµ ∂αθA .
α (2.1.1.36) This leads to the following proposal for a space time supersymmetric string action
S1 = − 1
4πα √
d2σ −γγ αβ Πµ Πβµ .
α (2.1.1.37) Note that in contrast to the previously discussed worldsheet supersymmetric string,
(2.1.1.37) consists only of bosons when looked at from a two dimensional point of view.
The action (2.1.1.37) is invariant under global target space supersymmetry, i.e. Lorentz 2. Worldsheet Actions 11 transformations plus the supersymmetry transformations (2.1.1.33) – (2.1.1.35). From
the worldsheet perspective we have reparametrization invariance and Weyl invariance
(2.1.1.5). This is again enough to ﬁx the worldsheet metric γαβ = ηαβ (cf (2.1.1.6)).
The resulting action will exhibit conformal coordinate transformations (2.1.1.9) as
residual symmetries. The energy momentum tensor ((2.1.1.4) with ∂α X µ replaced by
µ Πα (2.1.1.36)) is again traceless. Like in section 2.1.1.1, the vanishing of the energy momentum tensor gives two constraints. We have seen that in the nonsupersymmetric
case ﬁxing conformal coordinate transformations and solving the constraints leaves
eﬀectively d − 2 (transversal) bosonic directions.6 In order for the target space super symmetry not to be spoiled in this process, we would like to reduce the number of
fermionic directions θA by a factor of
2 [d−2]
2 2 [d]
2 = 1
2 simultaneously. So, we need an additional local symmetry whose gauge ﬁxing will
remove half of the fermions θA . The symmetry we are looking for is known as κ
symmetry. It exists only in special circumstances. First of all, the number of supersymmetries should not exceed N = 2 (i.e. A = 1, 2). Then, adding a further term
S2 = 1
2πα
+ d2σ −i αβ ¯
¯
∂ X µ θ1 Γµ ∂β θ1 − θ2 Γµ ∂β θ2 ¯
θ Γ ∂α θ1 θ2 Γµ ∂β θ2 αβ ¯1 µ (2.1.1.38) to (2.1.1.37) results in a κ symmetric action. (We will give the explicit transformations
below.) In (2.1.1.38) αβ denotes the two dimensional Levi Civita symbol. If one is
interested in less than N = 2 one can just put the corresponding θA to zero. The
requirement that adding S2 to the action does not spoil supersymmetry (2.1.1.33) –
(2.1.1.35), leads to further constraints,
(i)
d = 3 and θ is Majorana
(ii) d = 4 and θ is Majorana or Weyl
(iii) d = 6 and θ is Weyl
(iv ) d = 10 and θ is MajoranaWeyl.
It remains to give the above mentioned κ symmetry transformations explicitly.
By adding S1 and S2 one observes that the kinetic terms for the θ’s (terms with one
derivative acting on a fermion) contain the following projection operators
αβ
P± =
6 1
2 αβ γ αβ ± √
−γ . (2.1.1.39) Since the ﬁeld equations are diﬀerent for (2.1.1.37) the details of the discussion in the bosonic case
will change. The above frame just gives a rough motivation for a modiﬁcation of (2.1.1.37) carried
out below. 2. Worldsheet Actions 12 The transformation parameter for the additional local symmetry is called κA . It is a
α
spinor from the target space perspective and in addition a worldsheet vector subject
to the following constraints
αβ
κ1α = P− κ1 ,
β κ 2α = αβ
P+ κ2 ,
β (2.1.1.40)
(2.1.1.41)
(2.1.1.42) where the worldsheet indices α, β are raised and lowered with respect to the worldsheet
metric γαβ . Now, we are ready to write down the κ transformations,
δθA = 2iΓµ Παµ κAα ,
¯
δX µ = iθA Γµ δθA ,
√
√
αγ
αγ
δ
= −16 −γ P− κ1β ∂γ θ1 + P+ κ2β ∂γ θ2 .
−γγ αβ
¯
¯ (2.1.1.43)
(2.1.1.44)
(2.1.1.45) For a proof that these transformations leave S1 + S2 indeed invariant we refer to[222]
for example.
Once we have established that the number of local symmetries is correct, we can
now proceed to employ those symmetries and reduce the number of degrees of freedom
by gauge ﬁxing. We will go to the light cone gauge in the following. Here, we will
discuss only the most interesting case of d = 10. As usual we use reparametrization
and Weyl invariance to ﬁx γαβ = ηαβ . We can ﬁx κ symmetry (2.1.1.43)–(2.1.1.45) by
the choice
Γ+ θ1 = Γ+ θ2 = 0, (2.1.1.46) 1
Γ± = √ Γ0 ± Γ9 .
2 (2.1.1.47) where This sets half of the components of θ to zero. With the κ ﬁxing condition (2.1.1.46)
the equations of motion for X + and X i (i = 2, . . . , d − 1) turn out to be free ﬁeld equations (cf (2.1.1.11)). The reason for this can be easily seen as follows. After
A
imposing (2.1.1.46), out of the fermionic terms only those containing θ¯ Γ− θA remain
in the action S1 + S2 . Especially, the terms fourth order in θA have gone. The above
mentioned terms with Γ− couple to ∂α X + , and hence they will only have inﬂuence
on the X − equation (obtained by taking the variation of the action with respect to
X +). Thus we can again ﬁx the conformal coordinate transformations by the choice
(2.1.1.13). The X − direction is then ﬁxed (up to a constant) by imposing the constraint
of vanishing energy momentum tensor. Since the coupling of bosons and fermions is 2. Quantization of the fundamental string 13 reduced to a coupling to ∂α X +, there is just a constant p+ in front of the free kinetic
terms of the fermions.
In the lightcone gauge described above the target space symmetry has been ﬁxed
up to the subgroup SO(8), where the X i and the θA transform in eight dimensional
representations.7 For SO(8) there are three inequivalent eight dimensional representations, called 8v , 8s , and 8c . The group indices are chosen as i, j, k for the 8v , a, b, c for
the 8s , and a, b, c for the 8c . In particular, X i transforms in the vector representation
˙˙˙
8v . For the target space spinors we can choose either 8s or 8c . Absorbing also the
constant in front of the kinetic terms in a ﬁeld redeﬁnition we specify this choice by
the following notation
p + θ 1 → S 1a
p + θ 2 → S 2a ˙
or S 1a (2.1.1.48) ˙
or S 2a. (2.1.1.49) Essentially, we have here two diﬀerent cases: we take the same SO(8) representation
for both θ’s or we take them mutually diﬀerent. The ﬁrst option results in type IIB
theory whereas the second one leads to type IIA.
So, the gauge ﬁxing procedure simpliﬁes the theory substantially. The equations of
motion for the remaining degrees of freedom are just free ﬁeld equations. For example
for the type IIB theory they read,
∂+ ∂− X i = 0,
∂+ S 1a = 0, ∂− S 2a = 0. (2.1.1.50)
(2.1.1.51)
(2.1.1.52) They look almost equivalent to the equations of motion one obtains from the worldsheet supersymmetric action (2.1.1.15) after eliminating the ± directions by the light
cone gauge. Especially, (2.1.1.51) and (2.1.1.52) have the form of two dimensional
Dirac equations where S 1 and S 2 appear as 2d MajoranaWeyl spinors. An important
diﬀerence is however, that in (2.1.1.15) all worldsheet ﬁelds transform in the vector
representation of the target space subgroup SO(d − 2).
In the rest of this chapter we will focus only on the worldsheet supersymmetric
formulation. There, target space fermions will appear in the Hilbert space when quantizing the theory. We will come back to the Green Schwarz string only when discussing
type IIB strings living in a nontrivial target space (AdS5 × S 5) in section 4.3. 2.1.2 Quantization of the fundamental string 7
A MajoranaWeyl spinor in ten dimensions has 16 real components. Imposing (2.1.1.46) leaves
eight. 2. Quantization of the fundamental string
2.1.2.1 14 The closed bosonic string Our starting point is equation (2.1.1.11).
∂+ ∂− X i = 0. (2.1.2.1) Imposing periodicity under shifts of σ 1 by π we obtain the following general solutions8
µ
µ
X µ = XR σ − + XL σ + , (2.1.2.2) with
µ
XR =
µ
XL = 1µ 1µ− i
x + pσ +
2
2
2
1µ 1µ+ i
x + pσ +
2
2
2 n=0 n=0 1 µ −2inσ−
αe
,
nn (2.1.2.3) 1 µ −2inσ+
αe
˜
.
nn (2.1.2.4) Here, all σ α dependence is written out explicitly, i.e. xµ , pµ , αµ , and αµ are σ α
˜n
n
µ with the center of mass
independent operators. Classically, one can associate x
position, pµ with the center of mass momentum and αµ (αµ ) with the amplitude
n ˜n
of the n’th right moving (left moving) vibration mode of the string in xµ direction.
Reality of X µ imposes the relations
αµ† = αµ n and αµ† = αµ n .
˜n
˜−
n
− (2.1.2.5) We also deﬁne a zeroth vibration coeﬃcient via
µ µ ˜
α0 = α0 = 1µ
p.
2 (2.1.2.6) Since the canonical momentum is obtained by varying the action (2.1.1.8) with
˙
respect to X µ (where the dot means derivative with respect to τ ) we obtain the
following canonical quantization prescription. The equal time commutators are given
by
X µ (σ ) , X ν σ ˙
˙
= X µ (σ ) , X ν σ = 0, (2.1.2.7) and
˙
X µ (σ ) , X ν σ = −iπδ σ − σ η µν (2.1.2.8) where the delta function is a distribution on periodic functions. Formally it can be
assigned a Fourier series
1
δ (σ ) =
π ∞ e2ikσ . (2.1.2.9) k =−∞ 8
Frequently, we will put α = 1 . Since it is the only dimensionfull parameter (in the system with
2
= c = 1), it is easy to reinstall it when needed. 2. Quantization of the fundamental string 15 With this we can translate the canonical commutators (2.1.2.7) and (2.1.2.8) into
commutators of the Fourier coeﬃcients appearing in (2.1.2.3) and (2.1.2.4),
[pµ , xν ] = −iη µν , [αµ , αν ]
n
k (2.1.2.10)
µν = nδn+k η , (2.1.2.11) [˜ µ , αν ] = nδn+k η µν ,
αn ˜ k (2.1.2.12) where δn+k is shorthand for δn+k,0 . So far, we did not take into account the constraints
of vanishing energy momentum tensor (2.1.1.10). To do so we go again to the light
cone gauge (2.1.1.13), i.e. set
α+ = α+ = 0 for n = 0.
˜n
n (2.1.2.13) Now the constraint (2.1.1.10) can be used to eliminate X − (up to x− ), or alternatively
the α− and α− ,
˜n
n
p+ α− =
n
p+ α− =
˜n ∞
m=−∞
∞
m=−∞ : αi −m αi : −2aδn ,
n
m (2.1.2.14) : αi −m αi : −2aδn
˜n ˜ m (2.1.2.15) where a sum over repeated indices i from 2 to d − 1 is understood. The colon denotes
normal ordering to be speciﬁed below. We have parameterized the ordering ambiguity
by a constant a. (In principle one could have introduced two constants a, a. But this
˜
would lead to inconsistencies which we will not discuss here.) Equations (2.1.2.14) and
(2.1.2.15) are not to be read as operator identities but rather as conditions on physical
states which we will construct now. We choose the vacuum as an eigenstate of the pµ
pµ k = kµ k , (2.1.2.16) with kµ being an ordinary number. Further, we impose that the vacuum is annihilated
by half of the vibration modes,
αi k = αi k = 0 for n > 0.
˜n
n (2.1.2.17) The rest of the states can now be constructed by acting with a certain number of
αi n and αi n (n > 0) on the vacuum. But we still need to impose the constraint
˜−
−
(2.1.1.10). Coming back to (2.1.2.14) and (2.1.2.15) we can now specify what is meant
by the normal ordering. The αi (αi ) with the greater Fourier index k is written to
k ˜k
the right9. For n = 0 (2.1.2.14) and (2.1.2.15) just tell us how any α− or α− can be
˜n
n
9
E.g. for k > 0 this implies that : αi αi k := αi k αi , i.e. the annihilation operator acts ﬁrst on a
k−
−
k
state. 2. Quantization of the fundamental string 16 expressed in terms of the αi and αi . The nontrivial information is contained in the
˜l
k
n = 0 case. It is convenient to rewrite (2.1.2.14) and (2.1.2.15) for n = 0,
˜
2p+ p− − pi pi = 8(N − a) = 8(N − a), (2.1.2.18) where (doing the normal ordering explicitly)
N
˜
N =
= ∞
n=1
∞
n=1 αi n αi ,
−
n (2.1.2.19) αi n αi .
˜− ˜n (2.1.2.20) ˜
The N (N ) are number operators in the sense that they count the number of creation
˜
operators αi n (αi n ) acting on the vacuum. To be precise, the N (N ) eigenvalue of
˜−
−
a state is this number multiplied by the index n and summed over all diﬀerent kinds
of creation operators acting on the vacuum (for left and right movers separately).
Interpreting the pµ eigenvalue kµ as the momentum of a particle (2.1.2.18) looks like
a mass shell condition with the mass squared M 2 given by
˜
M 2 = 8(N − a) = 8(N − a). (2.1.2.21) The second equality in the above equation relates the allowed right moving creation
operators acting on the vacuum to the left moving ones. It is known as the level
matching condition.
For example, the ﬁrst excited state is
αi 1 αj 1 k .
− ˜− (2.1.2.22) By symmetrizing or antisymmetrizing with respect to i, j and splitting the symmetric
expression into a trace part and a traceless part one sees easily that the states (2.1.2.22)
form three irreducible representations of SO(d − 2). Since we have given the states
the interpretation of being particles living in the targetspace, these should correspond
to irreducible representations of the little group. Only when the above states are
massless the little group is SO(d − 2) (otherwise it is SO(d − 1)). Therefore, for
unbroken covariance with respect to the targetspace Lorentz transformation, the states
(2.1.2.22) must be massless. Comparing with (2.1.2.21) we deduce that the normal
ordering constant a must be one,
! a = 1. (2.1.2.23) In the following we are going to compute the normal ordering constant a. Requiring
agreement with (2.1.2.23) will give a condition on the dimension of the targetspace 2. Quantization of the fundamental string 17 to be 26. The following calculation may look at some points a bit dodgy when it
comes to computing the exact value of a. So, before starting we should note that the
compelling result will be that a depends on the targetspace dimension. The exact
numerics can be veriﬁed by other methods which we will not elaborate on here for
˜
the sake of briefness. We will consider only N since the calculation with N is a very
straightforward modiﬁcation (just put tildes everywhere). The initial assumption is
that naturally the ordering in quantum expressions would be symmetric, i.e.
1
N −a=
2 ∞
n=−∞,n=0 αi n αi .
−
n (2.1.2.24) By comparison with the deﬁnition of N (2.1.2.19) and using the commutation relations
(2.1.2.11) we ﬁnd
a=− d−2
2 ∞ n. (2.1.2.25) n=1 This expression needs to be regularized. A familiar method of assigning a ﬁnite number to the rhs of (2.1.2.25) is known as ‘zeta function regularization’. One possible
representation of the zeta function is
ζ (s) = ∞ n−s . (2.1.2.26) n=1 The above representation is valid for the real part of s being greater than one. The
zeta function, however, can be deﬁned also for complex s with negative real part. This
is done by analytic continuation. The way to make sense out of (2.1.2.25) is now to
replace the inﬁnite sum by the zeta function
a=− d−2
d−2
ζ (−1) =
.
2
24 (2.1.2.27) Comparing with (2.1.2.23) we see that we need to take
d = 26 (2.1.2.28) in order to preserve Lorentz invariance. This result can also be veriﬁed in a more rigid
way. Within the present approach one can check that a = 1 and d = 26 are needed
for the target space Lorentz algebra to close. In other approaches, one sees that the
Weyl symmetry becomes anomalous for d = 26.
˜
Since N and N are natural numbers we deduce from (2.1.2.21) that the mass
spectrum is an inﬁnite tower starting from M 2 = −8 = −4/α and going up in steps of
8 = 4/α . The presence of a tachyon (a state with negative mass square) is a problem. 2. Quantization of the fundamental string 18 Spin/ GRAVITON 4 2 0 4 8 M 2α Figure 2.1: Mass spectrum of the closed bosonic string
It shows that we have looked at the theory in an unstable vacuum. One possibility
that this is not complete nonsense could be that apart from the massterm the tachyon
potential receives higher order corrections (like e.g. a power of four term) with the
opposite sign. Then it would look rather like a Higgs ﬁeld than a tachyon, and one
would expect some phase transition (tachyon condensation) to occur such that the
ﬁnal theory is stable. For the moment, however, let us ignore this problem (it will not
occur in the supersymmetric theories to be studied next).
The massless particles are described by (2.1.2.22). The part symmetric in i, j and
traceless corresponds to a targetspace graviton. This is one of the most important
results in string theory. There is a graviton in the spectrum and hence string theory
can give meaning to the concept of quantum gravity. (Since Einstein gravity cannot
be quantized in a straightforward fashion there is a graviton only classically. This
corresponds to the gravitational wave solution of the Einstein equations. The particle
aspect of the graviton is missing without string theory.) The tracepart of (2.1.2.22) is
called dilaton whereas the piece antisymmetric in i, j is simply the antisymmetric tensor ﬁeld (commonly denoted with B ). A schematic summary of the particle spectrum
of the closed bosonic string is drawn in ﬁgure 2.1.
As a consistency check one may observe that the massive excitations ﬁt in SO(25)
representations, i.e. they form massive representations of the little group of the Lorentz
group in 26 dimensions.
As we have already mentioned, this theory contains a graviton, which is good since
it gives the prospect of quantizing gravity. On the other hand, there is the tachyon, at
best telling us that we are in the wrong vacuum. (There could be no stable vacuum
at all – for example if the tachyon had a run away potential.) Further, there are no 2. Quantization of the fundamental string 19 target space fermions in the spectrum. So, we would like to keep the graviton but to
get rid of the tachyon and add fermions. We will see that this goal can be achieved
by quantizing the supersymmetric theories.
2.1.2.2 Type II strings In this section we are going to quantize the (1,1) worldsheet supersymmetric string.
We will follow the lines of the previous section but need to add some new ingredients.
We start with the action (2.1.1.15). The equations of motion for the bosons X µ are
identical to the bosonic string. So, the mode expansion of the X µ is not altered and
given by (2.1.2.3) and (2.1.2.4). The equations of motion for the fermions are,
µ
∂− ψ+ = 0, µ
∂+ ψ− = 0. (2.1.2.29)
(2.1.2.30) Further, we need to discuss boundary conditions for the worldsheet fermions. Modulo
the equations of motion (2.1.2.29) and (2.1.2.30) the variation of the action (2.1.1.15)
with respect to the worldsheet fermions turns out to be10
i
µ
µπ
(2.1.2.31)
−ψ+µ δψ+ + ψ−µ δψ− σ=0 .
2π
µ
For the closed string we need to take the variation of ψ+ independent from the one of
µ
ψ− at the boundary (because we do not want the boundary condition to break part
of the supersymmetry (2.1.1.22) and (2.1.1.23)). Hence, the spinor components can
be either periodic or antiperiodic under shifts of σ by π . The ﬁrst option gives the
Ramond (R) sector. In the R sector the general solution to (2.1.2.29) and (2.1.2.30)
can be written in terms of the following mode expansion
µ
ψ− =
µ
ψ+ dµ e−2in(τ −σ) ,
n
n∈Z = (2.1.2.32) ˜
dµ e−2in(τ +σ) .
n (2.1.2.33) n∈Z The other option to solve the boundary condition is to take antiperiodic boundary
conditions. This is called the Neveu Schwarz (NS) sector. In the NS sector the general
solution to the equations of motion (2.1.2.29) and (2.1.2.30) reads11
µ
ψ− = bµ e−2ir(τ −σ) ,
r
˜µ e−2ir(τ +σ) ,
br µ
ψ+ =
10 (2.1.2.34)
(2.1.2.35) 1
r ∈Z+ 2 1
r ∈Z+ 2 Again we put α = 1 .
2
The reality (Majorana) condition on the worldsheet spinor components provides relations analogous to (2.1.2.5).
11 2. Quantization of the fundamental string 20 where now the sum is over half integer numbers (. . . , − 1 , 1 , 3 , . . . ).
222
For the bosons the canonical commutators are as given in (2.1.2.7), (2.1.2.8).
Hence, the oscillator modes satisfy again the algebra (2.1.2.10) – (2.1.2.12). Worldsheet fermions commute with worldsheet bosons. The canonical (equal time) anticommutators for the fermions are
µ
ν
ψ+ (σ ) , ψ+ σ µ
ν
= ψ− (σ ) , ψ− σ
µ
ν
ψ+ (σ ) , ψ− σ = πη µν δ σ − σ , (2.1.2.36) = 0. (2.1.2.37) For the Fourier modes this implies
br bs
{bµ , bν } = ˜µ , ˜ν
rs = η µν δr+s (2.1.2.38) in the NS sectors12, and
˜˜
{dµ , dν } = dµ , dν = η µν δm+n
mn
mn (2.1.2.39) in the R sectors. Like the bosonic Fourier modes these can be split into creation
operators with negative Fourier index, and annihilation operators with positive Fourier
index. What about zero Fourier index? For the NS sector fermions this does not occur.
The vacuum is always taken to be an eigenstate of the bosonic zero modes where the
eigenvalues are the target space momentum of the state. (This is exactly like in the
bosonic string discussed in the previous section.) The Ramond sector zero modes
form a target space Cliﬀord algebra (cf (2.1.2.39)). This means that the Ramond
sector states form a representation of the d dimensional Cliﬀord algebra, i.e. they are
target space spinors. We will come back to this later. Pairing left and right movers,
there are altogether four diﬀerent sectors to be discussed: NSNS, NSR, RNS, RR.
In the NSNS sector for example the left and right moving worldsheet fermions have
both antiperiodic boundary conditions. The vacuum in the NSNS sector is deﬁned
via (2.1.2.16), (2.1.2.17) and
bµ k = ˜µ k = 0 for r > 0.
br
r (2.1.2.40) We can build states out of this by acting with bosonic left and right moving creation
operators on it. Further, left and right moving fermionic creators from the NS sectors
can act on (2.1.2.40). We should also impose the constraints (2.1.1.24) – (2.1.1.27) on
those states. As before, we do so by going to the light cone gauge
α+ = α+ = b+ = ˜+ = 0.
˜n
br
n
r
12 one. (2.1.2.41) We say NS sectors and not NS sector because there are two of them: a left and a right moving 2. Quantization of the fundamental string 21 Then the constraints can be solved to eliminate the minus directions. The important
information is again in the zero mode of the minus direction. This reads (2.1.2.18)
˜
2p+ p− − pi pi = 8(NN S − aN S ) = 8(NN S − aN S ). (2.1.2.42) The expressions for the number operators are modiﬁed due to the presence of (NS
sector) worldsheet fermions
∞ NN S = n=1 αi n αi +
−
n ∞
1
r= 2 rbi r bi ,
−r (2.1.2.43) ˜
and the analogous expression for NN S . Its action on states is like in the bosonic
case (see discussion below (2.1.2.20)) taking into account the appearance of fermionic
creation operators. Again, we have put a so far undetermined normal ordering constant
in (2.1.2.42) and taken normal ordered expressions for the number operators. Now,
the ﬁrst excited state is
b
bi 1 ˜j 1 k .
−−
2 (2.1.2.44) 2 Its target space tensor structure is identical to the one of (2.1.2.22). In particular it
forms massless representations of the target space Lorentz symmetry. Thus, Lorentz
covariance implies that
aN S = 1
2 (2.1.2.45) should hold.
We compute now aN S by ﬁrst naturally assuming that a symmetrized expression
appears on the rhs of (2.1.2.42). This gives (see also (2.1.2.25))
aN S = − d−2
2 ∞ n+ n=1 d−2
2 ∞ r. (2.1.2.46) r= 1
2 We use again the zeta function regularization to make sense out of (2.1.2.46). For the
second sum the following formula proves useful
∞
n=0 (n + c) = ζ (−1, c) = − 1
6c2 − 6c + 1 .
12 (2.1.2.47) (Note, that splitting the lhs of (2.1.2.47) into ζ (−1)+ c + cζ (0) gives a diﬀerent (wrong)
result. This is because we understand the inﬁnite sum as an analytic continuation of
a ﬁnite one:
(n + a)−s with real part of s greater than one. For generic s the above 2. Quantization of the fundamental string 22 splitting is not possible.) Anyway, with the regularization prescription (2.1.2.47) we
get for (2.1.2.46)
aN S = d−2
.
16 (2.1.2.48) We conclude that the critical dimension for the (1, 1) worldsheet supersymmetric string
is
d = 10. (2.1.2.49) Like in the bosonic string there are more rigid calculations giving the same result.
The massless spectrum from the NSNS sector is identical to the massless spectrum
of the closed bosonic string. Again, we have a tachyon: the NSNS groundstate. Here,
however this can be consistently projected out. This is done by imposing the GSO
(GliozziScherkOlive) projection. To specify what this projection does in the NS sector
˜
we introduce fermion number operators F (F ) counting the number of worldsheet
fermionic NS right (left) handed creation operators acting on the vacuum. In addition,
˜
we assign to the right (left) handed NS vacuum an F (F ) eigenvalue of one13 . Now, the
GSO projection is carried out by multiplying states with the GSO projection operator
˜ PGSO = 1 + (−1)F 1 + (−1)F
.
2
2 (2.1.2.50) Obviously this does not change the ﬁrst excited state (2.1.2.44) but removes the tachyonic NSNS ground state. There are several reasons why this projection is consistent.
At tree level14 for example one may check that the particles which have been projected
out do not reappear as poles in scattering amplitudes. Imposing the GSO projection
becomes even more natural when looking at the one loop level. In the Euclidean
version this means that the worldsheet is a torus. Summing over all possible spin
structures (the periodicities of worldsheet fermions when going around the two cycles
of the torus) leads naturally to the appearance of (2.1.2.50) in the string partition
function [415] (see also [331]). The NSNS spectrum subject to the GSO projection
looks as follows. The number operator (2.1.2.43) is quantized in halfinteger steps.
The GSO projection removes half of the states, the groundstate, the ﬁrst massive
states, the third massive states and so on. The NSNS spectrum of the type II strings
is summarized in ﬁgure 2.2.
We have achieved our goal of removing the tachyon from the spectrum while keeping the graviton. We also want to have target space spinors. We will see that those
˜
This means that we can write F = 1 + r>0 bi r bi , and an analogous expression for F .
−r
The worldsheet has the topology of a cylinder, or a sphere when Wick rotated to the Euclidean
2d signature.
13 14 2. Quantization of the fundamental string 23 Spin/ GRAVITON 2 2 0 2 4 M 2α Figure 2.2: NSNS spectrum of the type II string. In comparison to ﬁgure 2.1 the
horizontal axis has been stretched by a factor of two.
come by including the R sector into the discussion. The most important issue to be
addressed here is the action of the zero modes on the R groundstate. By going to the
lightcone gauge, we can again eliminate the plus and minus (or the 0 and 1) directions leaving us with eight15 zero modes for the left and right moving sectors each.
We rearrange these modes into four complex modes
D1 = d2 + id3,
0
0 (2.1.2.51) id5,
0 (2.1.2.52) D3 = d6 + id7,
0
0 (2.1.2.53) D4 = d8 + id9.
0
0 (2.1.2.54) D2 = d4
0 + The only nonvanishing anticommutators for these new operators are (I = 1, . . . , 4;
no sum over I )
†
DI , DI = 2. (2.1.2.55) †
In particular, the DI and DI are nilpotent. We can now construct the right moving
R vacuum by starting with a state which is annihilated by all the DI ,16 DI −, −, −, − = 0 for all I. (2.1.2.56) †
Acting with a DI on the vacuum changes the I th minus into a plus, e.g.
†
D3 −, −, −, − = −, −, +, − .
15 (2.1.2.57) We use here the previous result that we need to have d = 10 in order to preserve target space
Lorentz invariance.
16
In this notation we suppress the eigenvalue k µ of the bosonic zero modes. 2. Quantization of the fundamental string 24 †
Acting once more with D3 will give zero. Acting with D3 on (2.1.2.57) will give back
(2.1.2.56) because of (2.1.2.55). Thus, we have a 24 = 16fold degenerate vacuum. This gives an on shell Majorana spinor in ten dimensions. For the left movers the
construction is analogous. (The above method to construct the state is actually an
option to construct (massless) spinor representations when the di are identiﬁed with
0
the target space Gamma matrices.) Without further motivation (which is given in the
books and reviews listed in section 6) we state how the GSO projection is performed
in the R sector. First, we deﬁne
(−1)F = 24 d2d3 d4d5d6 d7d8 d9 (−1)
00000000 n>0 di n di
n
− , (2.1.2.58) where the factor of 24 has been introduced such that (−)2F = 1, ensuring that
√
(2.1.2.59) deﬁnes projection operators. Note also that Γµ = 2dµ satisﬁes the canon0
ically normalized Cliﬀord algebra {Γµ , Γν } = 2η µν . For the groundstate this is just
the chirality operator (the product of all Gamma matrices) in the transverse eight
dimensional space. Now, we multiply the R states by one of the following projection
operators
±
PGSO = 1 ± (−1)F
2 (2.1.2.59) We perform the analogous construction in the left moving R sector. There are essentially two inequivalent options: we take the same sign in (2.1.2.59) for left and right
movers, or diﬀerent signs. Taking diﬀerent signs leads to type IIA strings whereas the
option with the same signs is called type IIB. Multiplying the R groundstate with one
of the operators (2.1.2.59) reduces the 16 dimensional Majorana spinor to an eight
dimensional Weyl spinor17.
To complete the discussion of the R sector we have to combine left and right movers,
i.e. to construct the NSR, RNS, and RR sector of the theory. Let us start with the
NSR sector. The mass shell condition (2.1.2.42) reads now
2p+ p− − pi pi = 8 NN S − 1
2 ˜
= 8NR, (2.1.2.60) where the number operator in the R sector is given as
NR = ∞
n=1 αi n αi
−
n + ∞
n=1 ndi n di ,
−n (2.1.2.61) and the analogous expression for the left movers. We have put the normal ordering
constant in the Ramond sector to zero. This can easily be justiﬁed by replacing the
17
The two diﬀerent choices in (2.1.2.59) give either the 8s or the 8c representation of SO(8) mentioned in section 2.1.1.3 2. Quantization of the fundamental string 25 half integer modded sum over r by an integer modded one in (2.1.2.46). Level matching
implies that the lowest allowed state in the NSR sector is massless and given by
bi 1 k ua
− (2.1.2.62) 2 where ua denotes the eight component MajoranaWeyl spinor comming from the R
ground states surviving the GSO projection. The 64 states contained in (2.1.2.62)
decompose into an eight dimensional and a 56 dimensional representation of the target
space little group SO(8). The 56 dimensional representation gives a gravitino of ﬁxed
chirality, whereas the eight dimensional one gives a dilatino of ﬁxed chirality.
The discussion of the RNS sector goes along the same line giving again a gravitino
and a dilatino either of opposite (to the NSR sector) chiralities corresponding to type
IIA theory, or of the same chiralities when the type IIB GSO projection is imposed.
Finally, in the RR sector the lowest state is obtained by combining the left with
the right moving vacuum. This state is massless due to the normal ordering constant
aR = 0. It has 64 components. The irreducible decompositions of the RR state
depend on whether we have imposed GSO conditions corresponding to type IIA or
type IIB. In the type IIA case the 64 states decompose into an eight dimensional
vector representation and a 56 dimensional representation. Thus in the type IIA
theory, the RR sector gives a massless U (1) oneform gauge potential Aµ and a threeform gauge potential Cµνρ . In the type IIB theory the 64 splits into a singlet, a 28 and
a 35 dimensional representation of SO(8). This corresponds to a “zeroform” Φ , a
∗
twoform Bµν , and a fourform gauge potential with selfdual ﬁeld strength Cµνρσ . The
particle content of the type II theories can be arranged in to N = 2 supermultiplets of
chiral (type IIB) or nonchiral (type IIA) ten dimensional supergravity. The (target
space bosons of the) massless spectrum of the type II strings is summarized in table
2.1.
2.1.2.3 The heterotic string Since the heterotic string is a bit out of the focus of the present review we will brieﬂy
state the results. The starting point is the action (2.1.1.32). Without the λA this
+
looks like the type II theories with the left handed worldsheet fermions removed.
Indeed, this part of the theory leads to the spectrum of the type II theories with
only the NS and R sector. The massless spectrum corresponds to N = 1 chiral
supergravity in ten dimensions. It corresponds to the states (the αi are the Fourier
˜n
i for the right moving fermions in the
coeﬃcients for the left moving bosons, and the br
NS sector)
αi 1 bj 1 k ,
˜− −
2 (2.1.2.63) 2. Quantization of the fundamental string 26 in the NS sector, and
αi 1 k uα
˜− (2.1.2.64) in the R sector, where we denoted again the GSO projected R vacuum with uα . The
above states must be massless since they form irreducible representations of SO(d − 2).
Focusing on the right moving sector we can deduce that the right moving normal
1
ordering constant must be 2 like in the type II case. Hence, the number of dimensions
(range of µ) is ten. As it stands the above spectrum leads to an anomalous theory.
But there is still the option of switching on the λA . Let us ﬁrst deduce the number
+ of additional directions (labeled by A) needed. In the sector where the vacuum is
non degenerate due to the presence of the λA , we know that we need the left moving
+
normal ordering constant to be one. (Otherwise the states (2.1.2.63) would not be
massless, but still form SO(d − 2) representations.) The vacuum does not receive
further degeneracy in the sector where all of the λ+ have antiperiodic boundary
A
conditions. In this sector the normal ordering constant is (see also (2.1.2.46)), the
label A stands for antiperiodic
aA =
˜ d−2 D
+,
24
48 (2.1.2.65) where we have called the number of additional directions D (A = 1, . . . , D). The
consistency condition aA = 1 tells us that there must be 32 additional directions,
˜
D = 32. (2.1.2.66) Let us ﬁrst discuss the simplest option, namely that all of the λA have always
+
identical boundary conditions, either periodic or antiperiodic. In the periodic sector
1
one easily computes that the normal ordering constant ˜P is negative (− 3 ). Hence,
a
there are no massless states in this sector. In the NS sector we ﬁnd in addition to
(2.1.2.63) the massless states (denoting with ˜A the Fourier coeﬃcients of λA in the
br
+
antiperiodic sector)
˜A 1 ˜B 1 bi 1 k .
b− b− −
2 2 (2.1.2.67) 2 Since the ˜A anticommute this is an antisymmetric 32 × 32 matrix. In addition it
b
is a target space vector (because of the index i). Therefore, the state (2.1.2.67) is an
SO(32) gauge ﬁeld. The corresponding R sector provides (after imposing the GSO
projection) fermions ﬁlling up an N = 1 supermultiplet in ten dimensions. Together,
with this SO(32) YangMills part the ten dimensional ﬁeld theory with the same
massless content is anomaly free. The GSO projection in the periodic sector is such
that only states with an even number of left moving fermionic creators survive. In the 2. Quantization of the fundamental string IIA # of Q’s
32 # of ψµ ’s
2 IIB 32 2 heterotic
E8 × E 8
heterotic
SO(32) 16 1 16 1 27 massless bosonic spectrum
NSNS Gµν , Bµν , Φ
RR
Aµ , Cµνρ
NSNS Gµν , Bµν , Φ
∗
RR
Cµνρσ , Bµν , Φ
Gµν , Bµν , Φ
Aa in adjoint of E8 × E8
µ
Gµν , Bµν , Φ
a in adjoint of SO (32)
Aµ Table 2.1: Consistent closed string theories in ten dimensions.
P sector it removes half of the groundstates (leaving only spinors of deﬁnite chirality
with respect to the internal space spanned by the A directions).
Another option is to group the λA into two groups of 16 directions. Then we would
+
naturally split the state (2.1.2.67) into three groups: (120, 1), (1, 120), and (16, 16),
depending on whether A and B in (2.1.2.67) are both in the ﬁrst half (1, . . . , 16),
both in the second half (17, . . . , 32), or one of them out of the ﬁrst half and the other
one out of the second half. So far, this gave only a rearrangement of those states.
But now we impose the GSO projection such that only states survive where an even
number of fermionic left moving creators act in each half separately. This removes the
(16, 16) combination. Further, when we split the range of indices into two groups of
16 each, there will be additional massless states. It is simple to check that in the sector
where half of the boundary conditions are periodic and the other half is antiperiodic
(the AP or PA sector), the left moving normal ordering constant vanishes. Hence, the
corresponding ground states give rise to massless ﬁelds, provided right moving creation
operators act such that level matching is satisﬁed. This gives (removing half of those
states by GSO projection) (128, 1) additional massless vectors from the PA sector,
and another (1, 128) from the AP sector. Together with the vectors from the AA
sector this gives an E8 × E8 YangMills ﬁeld. The R sector state ﬁlls in the fermions
needed for N = 1 supersymmetry in ten dimensions. This corresponds to the other
known N = 1 anomaly free ﬁeld theory.
The bosonic parts of the massless spectra of the consistent closed string theories in
ten dimensions is summarized in table 2.1. We have added the number of supercharges
Q from a target space perspective, and also the number of worldsheet supersymmetries
ψµ, in the NSR formulation. 2. Strings in nontrivial backgrounds 2.1.3 28 Strings in nontrivial backgrounds In the previous sections we have seen that all closed strings contain a graviton, a
dilaton, and an antisymmetric tensor ﬁeld in the massless sector. This is called the
universal sector. So far, we have studied the situation where the target space metric is
the Minkowski metric, the antisymmetric tensor has zero ﬁeld strength and the dilaton
is constant. In order to investigate what happens when we change the background,
we need to modify the action (2.1.1.3) as follows (this action is called the string sigma
model)
1
4πα
1
−
4π S= d2σ √ αβ
γγ Gµν (X ) ∂α X µ∂β X ν + i αβ √
d2 σ γ Φ (X ) R(2), Bµν (X ) ∂αX µ ∂β X ν
(2.1.3.1) where R(2) is the scalar curvature computed from γαβ . Throughout this section we will
consider a Euclidean worlsheet signature. Note, that the dilaton term does not contain
α . In general, the theory (2.1.3.1) cannot be quantized in an easy way. The best one
can do is to take a semiclassical approach. Since α enters like in ordinary ﬁeld
theories this will result in a perturbative expansion in α . The term with the dilaton
can be viewed as a ﬁrst order contribution in this expansion. Without this term,
(2.1.3.1) has again three local symmetries: diﬀeomorphisms and Weyl invariance. The
dilaton term breaks Weyl invariance in general. We will be interested in the question
under which circumstances Weyl invariance remains unbroken in the semiclassically
quantized theory. To answer this, ﬁrst note that Gµν , Bµν , and Φ can be viewed as
couplings from a two dimensional perspective. Weyl invariance in particular implies
global scale invariance. But scale invariance is related to vanishing beta functions in
ﬁeld theory. Thus, we will compute the beta functions of Gµν , Bµν and Φ as a power
series in α . However, there is a subtlety here. Under ﬁeld redeﬁnitions (inﬁnitesimal
shifts of X by χ [X ]) the couplings change according to
δGµν
δBµν = 2D(µχν ) ,
ρ = χ Hρµν + ∂µ Lν − ∂ν Lµ , δ Φ = χρ ∂ρ φ, (2.1.3.2)
(2.1.3.3)
(2.1.3.4) where we have deﬁned
Hρλκ = ∂ρ Bλκ + ∂λ Bκρ + ∂κ Bρλ (2.1.3.5) Lκ = χρ Bκρ . (2.1.3.6) and 2. Strings in nontrivial backgrounds 29 Expression (2.1.3.5) deﬁnes a ﬁeld strength corresponding to the B ﬁeld. It is invariant
under a U (1) transformation
δBµν = ∂[µ Vν ] , (2.1.3.7) with Vµ being an arbitrary target space vector. It is easy to check that also (2.1.3.1)
possesses the invariance (2.1.3.7). The symmetry (2.1.3.7) can be taken care of by
allowing for arbitrary Lµ in (2.1.3.3). Thus the couplings and hence the beta functions
are not unique. But actually we will be not just interested in vanishing beta functions.
This would ensure only global scale invariance. The requirement of Weyl invariance is
more strict and will ﬁx the arbitrariness.
In order to compute the beta functions, we need to ﬁx the worldsheet diﬀeomorphisms. We leave the explicit form of the ﬁxed metric γαβ unspeciﬁed. The gauge
ﬁxing procedure introduces ghosts, the diﬀeomorphism invariance is replaced by BRST
invariance. The ghost action depends only on the 2d geometry. Therefore, we expect
that the ghosts contribute only to the dilaton beta function. We will not treat them
explicitly but guess their contribution in the end of this section. The semiclassical ap¯
proach means that we start from some background string X µ satisfying the equations
of motion. We study the theory of the ﬂuctuations around this background string.
Instead of using the ﬂuctuation in the coordinate ﬁeld X µ we will take the tangent
¯
vector to the geodesic connecting the background value X µ with the actual value X µ .
This diﬀerence is supposed to be small in this approximation. In order to compute
the tangent vectors we connect the background value and the actual position of the
string by a geodesic. The line parameter t is chosen such that at t = 0 we are at the
background position and at t = 1 at the actual position. The geodesic equation is (the
dot denotes the derivative with respect to t),
˙˙
¨
λµ + Γµ λν λρ = 0
νρ (2.1.3.8) ¯
λµ (0) = X µ , λµ (1) = X µ . (2.1.3.9) and the boundary conditions are Note that the target space Christoﬀel connection Γµ depends on X µ. The ﬁrst nonνρ
trivial eﬀects should come from terms second order in the ﬂuctuations in the action.
(First order terms vanish when the background satisﬁes the equations of motion.) We
¯
call the tangent vector to the geodesic (at X µ )
˙
ξ µ = λµ (0) . (2.1.3.10) One can solve (2.1.3.8) iteratively leading to a power series in t,
1
1
¯
λµ (t) = X µ + ξ µ t − Γµ ξ ν ξ ρ t2 − Γµ ξ ν ξ ρ ξ κ t3 + . . . ,
2 νρ
3! νρκ (2.1.3.11) 2. Strings in nontrivial backgrounds 30 where
Γµ =
νρκ µ
ν Γρκ = ∂ν Γµ − Γλ Γµ − Γλ Γµ .
ρκ
νρ λκ
νκ ρλ (2.1.3.12) Further, we may choose local coordinates such that only the constant and the term
linear in t appears in (2.1.3.11) and all higher order terms vanish in a neighborhood
¯
of X µ . (This is done by spanning the local coordinate system by tangent vectors to
geodesics.) The corresponding coordinates are called Riemann normal coordinates. In
¯
these coordinates the Taylor expansion of the various terms in (2.1.3.1) around X µ
takes the following form (up to second order in the ﬂuctuations),
1
¯
¯
¯
∂α X µ = ∂α X µ + Dαξ µ + Rµ λκν X ξ λ ξ κ ∂α X ν ,
3
1
¯
¯
Gµν (X ) = Gµν X − Rµρνκ X ξ ρ ξ κ ,
3
1
¯
¯
¯
Bµν (X ) = Bµν X + Dρ Bµν X ξ ρ + DλDρ Bµν X ξ λ ξ ρ
2
1
1
¯
¯
− Rλρµκ Bλν X ξ ρ ξ κ + Rλρνκ Bλµ X ξ ρ ξ κ ,
6
6
1
¯
¯
¯
Φ (X ) = Φ X + Dµ Φ X ξ µ + Dµ Dν Φ X ξ µ ξ ν ,
2 (2.1.3.13)
(2.1.3.14) (2.1.3.15)
(2.1.3.16) where Dρ denotes the usual covariant derivative in target space, and Rµ νρσ is the
target space Riemann tensor
Rµ νρλ = ∂ρ Γµ − ∂λΓµ + Γω Γµ − Γω Γµ .
νρ
νρ ωλ
νλ ωρ
νλ (2.1.3.17) Note that in the Riemann normal coordinates the contributions quadratic in the
Christoﬀels vanish. Further, we have deﬁned
µ
¯
Dαξ µ = ∂α ξ µ + Γλν ξ λ ∂α X ν . (2.1.3.18) Collecting everything, one can expand the action (2.1.3.1) in a classical contribution
S0 and a contribution due to ﬂuctuations. There will be no part linear in ξ µ as long
¯
as X µ satisﬁes the equations of motion. The ﬁrst nontrivial part is quadratic in the
ξ µ . We denote it by
(2) (2) (2) S (2) = SG + SB + SΦ , (2.1.3.19) 2. Strings in nontrivial backgrounds 31 ¯
with (the background ﬁelds G, B and Φ are taken at X µ )
(2) SG =− √
d2σ γγ αβ (Gµν Dαξ µ Dβ ξ ν 1
4πα ¯
¯
+Rρµκν ∂α X µ ∂β X ν ξ ρ ξ κ ,
(2) SB (2) SΦ =− =− 1
4πα 1
4π d2σi αβ (2.1.3.20) ¯
∂α X ρ Hρµν ξ ν Dβ ξ µ 1
¯
¯
+ DλHρµν ξ λ ξ ρ ∂α X µ ∂β X ν
2
√
1
d2σ γR(2) Dµ Dν Φξ µ ξ ν .
2 (2.1.3.21)
(2.1.3.22) The next step is to redeﬁne the ﬁelds ξ µ in terms of a vielbein,
µ ξ µ = EA ξ A , (2.1.3.23) with
Gµν AB
= Eµ Eν ηAB , µ EA EµB = ηAB . (2.1.3.24)
(2.1.3.25) In what follows, capital latin indices will be raised and lowered with the Minkowski
metric. The normal coordinate expansion is useful not only to get the expressions
(2.1.3.20), (2.1.3.21), (2.1.3.22) in a covariant looking form. An important advantage
of this method is that the functional measure (in a path integral approach) for the
ξ A is the usual translation invariant measure. This will simplify the computation of
the partition function. In order to be able to do the ﬁeld redeﬁnition (2.1.3.23) in a
meaningfull way we have to ensure that the ﬂuctuations are parameterized by target
space vectors. The tangent vectors to geodesics connecting the background with the
actual value are a natural choice. Before writing down the action in terms of the ξ A ,
we will absorb the ﬁrst term in (2.1.3.21) in an additional connection in the kinetic
term (the ﬁrst term in (2.1.3.20)). That can be done by adding and subtracting a
term looking like
¯
¯
∂α X ρ∂ α X κHρ λ µ Hκλν ξ µ ξ ν .
We deﬁne the covariant derivative on ξ A by plugging (2.1.3.23) into (2.1.3.18) and
introducing an additional connection
Dα ξ A = Dα ξ A + i αβ
ν
¯A
√ ∂β X ρ Eµ Hρ µ ν EB ξ B ,
2 −γ (2.1.3.26) where Dαξ A corresponds to the contribution from (2.1.3.18). The part of the action
quadratic in ﬂuctuations ﬁnally takes the form
S (2) = − 1
4πα √
d2σ γ γ αβ Dα ξ A Dβ ξA + MAB ξ A ξ B , (2.1.3.27) 2. Strings in nontrivial backgrounds 32 where the potential is
αβ ¯
¯
¯
¯
MAB = γ αβ ∂α X µ∂β X ν GµνAB + i √ ∂α X µ ∂β X ν BµνAB + α R(2)FAB .
γ (2.1.3.28) The matrices G , B and F do not have an explicit dependence on the worldsheet
coordinates and are given by
GµνAB
BµνAB
FAB 1
ρκ
= EA EB Rρµκν − Hµ λ ρ Hνλκ ,
4
1
λρ
DλHρµν EA EB ,
=
2
1µν
=
E E Dµ Dν Φ.
2AB (2.1.3.29)
(2.1.3.30)
(2.1.3.31) Since the action (2.1.3.27) is quadratic in the ﬂuctuations, integrating over the ﬂuctuations will result in the determinant of an operator. For the general form of the
operator in (2.1.3.27) it is very covenient to use known formulæ from the heat kernel
technique. In the heat kernel approach the partition function
Z= Dξ A eiS (2) can be expressed as a formal sum[202, 83]
dt −Ot 1
e
=
t
2 1
log Z =
2 ∞
µ−2 dt
t ∞ n an t 2 −1 , (2.1.3.32) n=−2 where is a dimensionless UV cutoﬀ and µ is a mass scale introduced for dimensional
reasons. The symbol O stands for the operator whose determinant is of interest. We
rescale t by α such that O has mass dimension 2.18 In order to compute the beta
functions, we are interested in the logarithmically divergent piece, i.e. in a2. This can
be found in the literature[202, 83]
a2 = 1
4π √
d
A
d2 σ γ −MA + R(2) .
6 (2.1.3.33) The divergence can be cancelled by adding appropriate counterterms to the action.
This amounts to a replacement of the bare (inﬁnite) couplings Gµν , Bµν , Φ in the
18 The appearance of a power series in α is more obvious in a Feynman diagramatic treatment.
There, the propagator goes like α whereas vertices go like 1/α . This relates directly the order of α
in logarithmically divergent diagrams to the number of loops. The disadvantage of this approach is
that the discussion for a general worldsheet metric γ is more involved. Fixing γ to be the Minkowski
metric results in problems when computing the dilaton beta function since R(2) vanishes for this
choice. 2. Strings in nontrivial backgrounds 33 following way,
α
log µ2 /
2
α
ren
log µ2 /
Bµν = Bµν −
2
1
Φ = Φren − log µ2 /
2 Gµν = Gren −
µν GµνA A , (2.1.3.34) BµνA A , (2.1.3.35) − d + cg
+ α FA A ,
6 (2.1.3.36) where G , B and F are as deﬁned in (2.1.3.29), (2.1.3.30) and (2.1.3.31) but now in
terms of the renormalized couplings. Further, we have included a possible contribution
of the diﬀeomorphism ﬁxing ghosts. Their action depends only on the intrinsic two
dimensional geometry and neither on the embedding in the target space nor on the
form of the background ﬁelds Gµν , Bµν and Φ. Therefore, the ghosts can contribute
only a constant renormalization of the dilaton Φ which we have parameterized by cg
in (2.1.3.36). The beta functions can be computed by taking the derivative of the
renormalized couplings with respect to log µ using the µ independence of the bare
couplings. Up to order α this leads to (they are all expressed in terms of renormalized
quantities and we supress the corresponding superscript in the following)
(G
βµν ) = α 1
Rµν − Hµ λρ Hνλρ ,
4 αλ
D Hλµν ,
2
d + cg α 2
=−
+ D Φ.
6
2 (2.1.3.37) (B
βµν ) = (2.1.3.38) β (Φ) (2.1.3.39) Because of the ambiguities related to the ﬁeld rediﬁnitions (2.1.3.2) – (2.1.3.4) we
cannot just set the beta functions to zero but only deduce that the model is Weyl
invariant (in ﬁrst approximation) if
(G
¯(G
βµν ) = βµν ) + D(µMν ) = 0,
1
(B
¯(B
βµν ) = βµν ) + Hµν λ Mλ + ∂[µ Lν ] = 0,
2
1
¯
β (Φ) = β (Φ) + ∂µ Φ M µ = 0
2 (2.1.3.40)
(2.1.3.41)
(2.1.3.42) The vectors Mµ and Lµ are not ﬁxed by just checking for global scale invariance. In
order to compute them we would need to impose (local) Weyl invariance. This could
be done by computing the expectation value of the trace of the energy momentum tensor. Instead of doing so, we will choose a rather indirect way of ﬁxing the ambiguity.
¯
Implicitly, we will be using a theorem stating that β (Φ) is constant if (2.1.3.40) and
(2.1.3.41) are satisﬁed[111]. In other words this means that up to a constant contribution (2.1.3.42) should be an integrability condition for the other two equations. Before 2. Strings in nontrivial backgrounds 34 deriving such a condition we need to study of which form the vectors Mµ and Lµ could
be at the given order in α . We want to express a vector in terms of our background
ﬁelds Gµν , Φ and Bµν . The ﬁeld Bµν should enter only via the gauge invariant ﬁeld
strength Hµνλ (since we have performed partial integegrations in S (2) such that the
beta functions come out in a gauge invariant form). With this information it is easy
to check that the only option we have is (a is some constant)
Mµ = a∂µ Φ , Lµ = 0, (2.1.3.43) where we do not consider a gradient contribution in Lµ since this would not be relevant.
The next step is to take the divergence of (2.1.3.40). Using the Bianchi identity (i.e.
the vanishing divergence of the Einstein tensor), the identity
H ρλµ Dρ Hλµν = 1
Dν Hµλρ H µλρ ,
6 and equations (2.1.3.40) and (2.1.3.41) one obtains
Dν a2
α
a2
(∂ Φ)2
D Φ + H2 +
2
12
2α = 0, (2.1.3.44) where we have deﬁned
H 2 = Hρνλ H ρνλ . (2.1.3.45) On the other hand, equation (2.1.3.42) implies
Dν a
α2
D Φ + (∂ Φ)2
2
2 = 0. (2.1.3.46) Without the H 2 term, (2.1.3.44) and (2.1.3.46) would be the same. The H 2 term in
the dilaton beta function is actually missing in our computation since we took into
account only one loop contributions. Any counterterm in the action leading to an
order α contribution in the dilaton beta function should be linear in α . Since, at tree
1
level the B ﬁeld enters with a factor α , the H 2 term in β (Φ) corresponds to a two loop
contribution. In our implicit approach we obtained this term (and in fact all order α
terms in β (Φ) ) without doing a two loop analysis. We were not able to ﬁx the value of the constant a, however. This is because
it could be absorbed in a rescaling of the ﬁeld Φ. But this would change the ratio
of the constant contribution to the dilaton beta function to the other contributions.
Therefore, a is not arbitrary. The constant a can be ﬁxed for example by studying
models with trivial metric and B ﬁeld and a linear dilaton. These models are much
easier to treat than the generic one. The result is
a = 2α . (2.1.3.47) 2. Strings in nontrivial backgrounds 35 Let us discuss the case of trivial metric and vanishing B ﬁeld a bit further. For a
¯
¯
linear dilaton, the β (G) and β (B ) vanish identically. According to the previously stated
¯
theorem (and to our result) the β (Φ) function is constant in this case. Models with
¯
that feature are known as conformal ﬁeld theories. The constant dilaton β function
is related to an anomaly of the transformation of the energy momentum tensor under
conformal coordinate changes (while keeping the worldsheet metric ﬁxed). If we ﬁx
the worldsheet metric to be the Minkowski metric the anomalous transformation of
the energy momentum tensor with respect to (2.1.1.9) reads
˜˜ ˜
Tσ+ σ+ = dσ +
dσ +
˜ 2 Tσ+ σ+ σ + + c
S σ+, σ+
˜
12 (2.1.3.48) where the second term denotes the Schwarz derivative
S (w, z ) = zz − 3
2 (z
2 (z ) )2 , (2.1.3.49) where z is a function of w and the primes denote derivatives. The Schwarz derivative
has the following chain rule
S (w (v (z )) , z ) = ∂v
∂w 2 S (v, z ) + S (w, v ) . (2.1.3.50) The transformation law (2.1.3.48) is the most general possibility such that associativity holds. An analogous consideration applies to T−− . Now, from (2.1.3.48) one can deduce part of the operator product expansion (OPE) of two energy momentum
tensors. To this end, one considers inﬁnitesimal transformations and uses the fact that
they are generated by T++ . One obtains the following OPE
Tσ+ σ+ σ + Tσ+ σ+ σ +
˜
˜˜ = c/2
2
+
T + + σ+
+ )4
+ − σ + )2 σ σ
−σ
(˜
σ
1
(2.1.3.51)
∂ + T + + σ+ + . . . ,
++
σ − σ+ σ σ σ
˜ (˜ +
σ where the dots stand for terms which are regular for σ + = σ + . For linear dilaton
˜
19, leading to (2.1.3.47).
backgrounds this OPE can easily be computed directly It remains to ﬁx the contribution coming from the gauge ﬁxing ghosts cg . This
can of course be calculated directly[375, 376]. Here, we will guess it correctly, instead.
From our discussion of the quantized bosonic string in the light cone gauge in 2.1.2 we
remember that the classical Lorentz covariance was preserved in d = 26. Comparing
with (2.1.3.48) we observe that our gauge ﬁxing procedure was justiﬁed only if c = 0.
Since, we did not have a linear dilaton background there, this can happen only if
cg = −26. (2.1.3.52) 19
One should ﬁrst compute Tαβ by varying the action with respect to γαβ , and gauge ﬁx γαβ = ηαβ
afterwards. 2. Perturbative expansion and eﬀective actions 36 Equation (2.1.3.52) can be conﬁrmed by an explicit computation (which can also be
viewed as an alternative way of deriving the critical dimension).
Given the fact, that a linear dilaton contributes to c, one may want to go directly to
d = 4 by switching on a linear dilaton. One obvious problem with this is however that
target space Lorentz covariance is broken explicitly – there is a distinguished direction
in which the dilaton derivative points. The more useful way of getting away from a
26 dimensional target space is to replace 22 of the string coordinates by a conformal
ﬁeld theory with central charge d → c = 22.
To summarize, up to order α the action (2.1.3.1) is Weyl invariant provided that
the following set of equations holds,
1
Rµν − Hµρλ Hν ρλ + 2Dµ ∂ν Φ = 0,
4
1
− DλHλµν + Hλµν Dλ Φ = 0,
2
1
α
1
2
(d − 26) − α D Φ + α (∂ Φ)2 − H 2 = 0.
6
2
24 2.1.4 (2.1.3.53)
(2.1.3.54)
(2.1.3.55) Perturbative expansion and eﬀective actions In the previous section we have seen that imposing Weyl invariance provides us with
constraints on the background in which the string propagates. These constraints can
be viewed as equations of motion for the background ﬁelds. Lifting those up to an
action would then yield an eﬀective ﬁeld theory description for the string theory. We
have discussed only the bosonic string, but an extension to the superstring is possible.
It may however be problematic. In the NSR formalism it is for example not possible
to include terms into the string sigma model which would correspond to nontrivial
RR backgrounds. Therefore, we will sketch an alternative method of computing an
eﬀective action here. We will not present any explicit calculations but just describe the
strategy. Starting from the spectrum and the amount of supersymmetries belonging to
a certain string theory one can write down a general ansatz for an eﬀective ﬁeld theory
action of the string excitation modes. This ansatz can be further ﬁxed by comparing
scattering amplitudes computed from the eﬀective description to amplitudes obtained
from a string computation. The string amplitudes can be described in a diagramatic
fashion as depicted in ﬁgure 2.3.
The external four legs (hoses) correspond to the two incoming particles scattering
into two outgoing particles. The expansion is in terms of the number of holes (the
genus) of the worldsheet. The ﬁrst diagram in 2.3 correponds to two incoming strings
joining into one string which in turn splits into two outgoing strings. In that sense it
contains two vertices. Analogously the second diagram contains four vertices and so
on. Assigning to each vertex one power of the string coupling gs , this gives a formal 2. Perturbative expansion and eﬀective actions + 37 +... Figure 2.3: Perturbative expansion of the four point function in a string computation
power series
A= ∞
n=0 2
gs n+2 A(n) . (2.1.4.1) This power series can be truncated after the ﬁrst contributions as long as gs
1. It
remains to specify what gs is. To this end, we ﬁrst observe that the power of gs in the
expansion terms in (2.1.4.1) is nothing but minus the Euler number of a worldsheet
with n handles and four boundaries. From (2.1.3.1) we know that the dilaton Φ
couples to the Euler density of the worldsheet. This follows immediately from the
GaussBonnet theorem
1
4π √
d2 σ γR(2) = χ ≡ 2 (1 − n) − b, (2.1.4.2) where n is the number of handles and b is the number of boundaries of the two
dimensional worldsheet. (The calculation of the scattering amplitudes is performed
after the worldsheet signature has been Wick rotated to the Euclidian one.) Thus, one
can identify
gs = e Φ , (2.1.4.3) where Φ denotes a constant vacuum expectation value (VEV) of the dilaton. (Remember from the previous section that a constant contribution to Φ was not ﬁxed
by the conformal invariance conditions. This is true for all string theories as can be
easily seen by noticing that a constant shift in Φ shifts the action (2.1.3.1) by a constant.) Therefore, the string coupling is an arbitrary parameter in the perturbative
approach to string theory. It is only restricted by the consistency requirement that
the perturbative expansion in ﬁgure 2.3 should not break down, i.e. gs
1.
There is also a second approximation in the computation of the scattering amplitudes. Since the massive string states have masses of the order of the Planck mass,
they are “integrated out”. This means that we are interested in eﬀects below the
Planck scale where those ﬁelds do not propagate. The eﬀective ﬁeld theory actions 2. Perturbative expansion and eﬀective actions 38 contain only the massless modes. For consistency, one should then restrict to processes
where the momentum transfer is p2
1/α .
The (bosonic part of the) eﬀective Lagrangians with at most two derivatives and
only the massless ﬁelds turn out to be
S = Suniv + Smodel , (2.1.4.4) where Suniv does not depend on which of the superstring theories we are looking at,
and Smodel is model dependent. The universal sector has as bosonic ﬁelds the metric,
the dilaton, and the B ﬁeld. The corresponding action is
Suniv = 1
2κ2 √
1
d10x −Ge−2Φ R + 4 (∂ Φ)2 − H 2 ,
12 (2.1.4.5) where κ2 ∼ (α )4 is the ten dimensional gravitational constant. Note that the set of equations of motion obtained from this action coincides with the conformal invariance conditions (2.1.3.53)– (2.1.3.55), with the diﬀerence that for the superstring the
constant contribution in the dilaton equation vanishes for d = 10.
For type II strings there are additional contributions giving the kinetic terms for
the RR gauge forms and ChernSimons interactions,
II
Smodel = − 1
2κ2 d10x
p 1
F2 ,
2(p + 2)! p+2 (2.1.4.6) where Fp+2 is the ﬁeldstrength of a p + 1 form RR gauge ﬁeld (plus –in some cases–
additional contributions which we will discuss later),
Fp+2 = dAp+1 + . . . . (2.1.4.7) The number p is the spatial extension of an object which couples electrically to the
p + 2 form gauge ﬁeld. In the worldvolume action of the corresponding p dimensional
object this coupling is
dp+1 σ i α1 ···αp+1 ∂α1 X µ1 · · · ∂αp+1 X µp+1 Aµ1 ···µp+1 . (2.1.4.8) For a point particle (p = 0) the above expression reads for example
i dτ dX µ
Aµ .
dτ From expression (2.1.3.1) we observe that the fundamental string is electrically charged
under the NSNS B ﬁeld. We will meet objects which are charged under the RR gauge
forms when discussing Dbranes in section 2.3. For the type IIA theory we have 2. Perturbative expansion and eﬀective actions 39 p = 0, 2, 4. Alternatively we could replace the ﬁeld strength in (2.1.4.6) by its Hodge
dual20
F8−p = Fp+2 . (2.1.4.9) In the type IIA theory the deﬁnition (2.1.4.7) is modiﬁed for the four form ﬁeld strength
F4 = dA3 + A1 ∧ H, (2.1.4.10) leading to a non standard Bianchi identity for the four form ﬁeld strength
dF4 = F2 ∧ H. (2.1.4.11) Finally, the ChernSimons interaction for type IIA is
IIA
SCS = − 1
8κ2 F4 ∧ F4 ∧ B. (2.1.4.12) For type IIB theories one has p = −1, 1, 3. For p = −1 the gauge form is a scalar,
which is called axion. The object which is electrically charged under this zero form is
localized in space and time. This is an instanton. The deﬁnition of the ﬁeld strength
(2.1.4.7) receives further contributions for p = 1 and p = 3
F3 = dA2 − A0 ∧ H,
1
1
F5 = dA4 − √ A2 ∧ H + √ B ∧ F3 .
3
3 (2.1.4.13)
(2.1.4.14) The ChernSimons interaction for the type IIB theory is
IIB
SCS = − 9
4κ2 A4 ∧ H ∧ F3 . (2.1.4.15) The ﬁve form ﬁeld strength has to be selfdual. This is not encoded in the action
(2.1.4.6) but has to be added as an additional constraint,
F5 = F5 . (2.1.4.16) In the heterotic string we have gauge ﬁelds transforming in the adjoint of SO(32)
or E8 × E8. Their ﬁeld strength is deﬁned as (we assign mass dimension one to the
√
gauge ﬁelds A – this is related to a α rescaling of A)
F = dA + A ∧ A.
20 (2.1.4.17) This can be done by adding the Bianchi identity dFp+2 = d (· · · ) with a Lagrange multiplier to
the action and integrating out Ap+1 . Because of covariance the Lagrange multiplier is a 7 − p form
and its ﬁeld strength is an 8 − p form. 2. Perturbative expansion and eﬀective actions 40 The deﬁnition of H in (2.1.4.5) needs to be modiﬁed21
H = dB − α
(ωY − ωL ) .
4 (2.1.4.18) The YangMills ChernSimons form ωY is
2
ωY = trA ∧ dA + trA ∧ A ∧ A,
3 (2.1.4.19) where A is the gauge connection of either E8 × E8 or SO(32). The modiﬁcation
(2.1.4.18) implies that the B ﬁeld transforms under gauge transformations and under
local Lorentz rotations in a nontrivial way such that H is gauge invariant. The
YangMills ChernSimons form has the property that its exterior derivative gives the
instanton density (in a four dimensional subspace with Euclidean signature),
dωY = trF ∧ F. (2.1.4.20) The Lorentz ChernSimons form is constructed from the spin connection ω ,
2
dωL = trω ∧ dω + trω ∧ ω ∧ ω.
3 (2.1.4.21) If its exterior derivative takes values only on a four dimensional submanifold with
Euclidean signature it corresponds to the Euler density of that manifold,
dωL = trR ∧ R. (2.1.4.22) If we take the ten dimensional geometry to consist of a direct product of a six dimensional non compact and a four dimensional compact space (with Euclidean signature)
the modiﬁcation (2.1.4.18) implies restrictions on the allowed gauge bundles on the
four dimensional compact space. The integration of dH over a compact space should
vanish. It follows that the Euler number of this space must be equal to the instanton
number of the gauge bundle.
In addition to the universal piece (2.1.4.5), the heterotic action contains a gauge
kinetic term and also the GreenSchwarz term which ensures anomaly cancellation
heterotic
Smodel = Sgauge + SGS , (2.1.4.23) with
Sgauge = −
21 1
2κ2 d10xe−2Φ α
trF 2 ,
8 (2.1.4.24) We present the eﬀective action for the heterotic string just for completeness, more details on
diﬀerential geometry and anomaly cancelation in the context of the eﬀective heterotic theory can be
found e.g. in [223]. 2. Perturbative expansion and eﬀective actions 41 where again, the trace is taken over the gauge group (E8 × E8 or SO(32)). The
GreenSchwarz term is
SGS = 8πi
α B ∧ X8 , (2.1.4.25) with (here, a power is meant with respect to the wedge product, e.g. F 4 ≡ F ∧F ∧F ∧F )
X8 = 111
2 (2π )6 48 5
1
trF 2
trF 4 −
4
8 2 1
1
1
− trF 2 trR2 + trR4 +
trR2
8
8
32 2 (2.1.4.26)
To close this section on eﬀective actions we identify the diﬀerent contributions
with the worldsheet topologies they correspond to. First, we observe that all the
terms appearing in the eﬀective actions are of a structure such that they contain some
power of eΦ times a factor which is invariant under constant shifts in Φ. In (2.1.4.3)
we have identiﬁed the string coupling as a constant part of eΦ . Thus, the leading term
in the perturbative expansion in ﬁgure 2.3 enters the eﬀective action accompanied
with a factor of e−2Φ . These are all terms in (2.1.4.5) and the gauge kinetic term
in the heterotic theory (2.1.4.24). One may be tempted to interpret the other terms
(containing no e−2Φ factor) as one loop contributions. This is, however, misleading.
In order to simplify the Bianchi identities for the RR gauge forms we have rescaled
the RR gauge potentials by eΦ . Undoing this rescaling means that for example the
RR form F2 receives a further contribution
A1 = e−Φ A1 −→ F2 = e−Φ dA1 − dΦ ∧ A1 ≡ e−Φ F2 , (2.1.4.27) and similar relations for the other RR ﬁeld strengths. (If the terms denoted by dots in
(2.1.4.7) contain other RR ﬁeld strengths additional Φ derivatives will be picked up.
But no relative power of eΦ will appear, since those terms always contain one RR ﬁeld
strength or potential and an NSNS ﬁeld strength or potential. The NSNS ﬁelds are
not rescaled.) After this rescaling all terms in the type II thoeries are of the structure
e−2Φ (invariant under Φ → Φ + constant) .
Since the rescaled (primed) ﬁelds correspond to the actual string excitations, the
eﬀective type II actions given here contain only tree level contributions.
2
As (implicitly) stated above, one loop contributions are multiplied by gs and hence
enter the eﬀective action with a factor of e−2Φ+2Φ = 1. In the type II examples, we
have seen that due to ﬁeld redeﬁnitions this correspondence may be changed. In
the heterotic case, however, there is no ﬁeld redeﬁnition such that all the terms in
the eﬀective action are multiplied by the same power of eΦ . Indeed, the appearance 2. Toroidal Compactiﬁcation and Tduality 42 of the GreenSchwarz term corresponds to a torus amplitude from the string theory
perspective. We excluded also higher orders in α which would lead to higher derivative
terms and contributions with massive string excitations.22 As long as the string scale
is much shorter (in length) than the scale of the process we are interested in those
terms can be neglected. 2.1.5 Toroidal Compactiﬁcation and Tduality In the previous sections we argued that perturbative superstring theories are consistent
provided that the target space is ten dimensional. As it stands, this cannot describe
our observable (four dimensional) world. At the end of section (2.1.3), we sketched as
a possible resolution to this problem the option to replace six of the target space dimensions by a conformal ﬁeld theory with the desired central charge. One simple way
to do so, is to replace a six dimensional subspace of the ten dimensional Minkowski
space by a compact manifold. The coordinates of that compact manifold should belong to a conformal ﬁeld theory with a consistent central charge. This restricts the
set of possible compactiﬁcations. The easiest option is to compactify the additional
directions on circles (by periodic identiﬁcation of the corresponding coordinates). This
clearly does not change the central charge contribution of those directions, since the
central charge depends only on local features of the target space.
2.1.5.1 KaluzaKlein compactiﬁcation of a scalar ﬁeld Before discussing some details of torus compactiﬁcations of string theories we recall
the KaluzaKlein compactiﬁcation of a free massless scalar ﬁeld. This will enable us
to appreciate new “stringy” features which we will study afterwards. Let us start with
a free massless scalar living in a ﬁve dimensional Minkowski space. We label the ﬁrst
four coordinates with a greek index µ = 0, . . . , 3 and call the ﬁfth direction y . The
ﬁve dimensional ﬁeld equation for the scalar ϕ is
2
η µν ∂µ ∂ν + ∂y ϕ (xµ , y ) = 0. (2.1.5.1) Now, we compactify the ﬁfth direction on a circle of radius R
y ≡ y + 2πR. (2.1.5.2) Solutions to (2.1.5.1) have to respect the periodicity (2.1.5.2). Therefore, the y dependent part of ϕ can be expanded into a Fourier series of periodic functions. Focusing
22 An exception is the ωL correction in (2.1.4.18) and the Green Schwarz term. They can be deduced
by using supersymmetry and anomaly cancellation. 2. Toroidal Compactiﬁcation and Tduality 43 on the nth Fourier mode, we ﬁnd
n ϕn (xµ , y ) = ϕn (xµ ) ei R y , (2.1.5.3) with integer n, i.e. the momentum in the ﬁfth direction is quantized. Plugging (2.1.5.3)
back into (2.1.5.1) leads to
η µν ∂µ ∂ν − m2 ϕn (xµ ) = 0,
n (2.1.5.4) with
n
,
R mn = (2.1.5.5) i.e. the nth Fourier mode leads in the eﬀective four dimensional description to a KleinGordon ﬁeld with mass (2.1.5.5). Since the general solution of (2.1.5.1) is a superposition of all Fourier modes the four dimensional description contains an inﬁnite
KaluzaKlein tower of massive four dimensional ﬁelds (depending only on the xµ ).
There are two limits to be discussed. The decompactiﬁcation limit is R → ∞. In this case all the KaluzaKlein masses (2.1.5.5) vanish. The four dimensional description breaks down. The other limit is R → 0 (or the compactiﬁcation radius becomes
much shorter than the experimental distance resolution). In this case, the KK masses
(2.1.5.5) become inﬁnite except for n = 0. Only the massless mode survives and no
trace from the ﬁfth dimension is left. This picture is very diﬀerent in string theories
as we will see now.
2.1.5.2 The bosonic string on a circle Even though the bosonic string is inconsistent because it contains a tachyon, we will
ﬁrst study the compactiﬁcation of the bosonic string on a circle. The essential stringy
properties will be visible in this toy model. We compactify the 26th coordinate (the
25th spatial direction),
x25 ≡ x25 + 2πR. (2.1.5.6) In the point particle limit string theory is just quantum mechanics of a free relativistic
25
particle. The plane wave solution contains the factor eip25 x where p25 is the center of
mass momentum in the 25th direction. This wave function should be periodic under
(2.1.5.6). This leads to a quantization condition for the center of mass momentum in
the compact direction
p25 = n
,
R (2.1.5.7) 2. Toroidal Compactiﬁcation and Tduality 44 with integer n (the momentum number). So far, everything is analogous to the free
scalar ﬁeld discussed above. The new stringy property arises by observing that the
string can wind around the compact direction. Technically, this means that the periodic boundary condition for the closed string is modiﬁed
X 25 (τ, σ + π ) = X 25 (τ, σ ) + 2πmR, (2.1.5.8) where the integer m denotes the winding number. With this ingredients the mode
expansions (2.1.2.3) and (2.1.2.4) are
25
XR =
25
XL = n
i
1 25
x+
− mR σ − +
2
2R
2
1 25
i
n
x+
+ mR σ + +
2
2R
2 k =0 k =0 1 25 −2ikσ−
αe
,
kk (2.1.5.9) 1 25 −2ikσ+
αe
˜
.
kk (2.1.5.10) Taking into account the compact direction, the mass shell condition has to be modiﬁed
in a straightforward way,
24
µ=0 pµ pµ = −M 2 . (2.1.5.11) Comparison with the constraints T++ = T−− = 0 (2.1.1.10) gives
M2 = 4 n
− mR
2R 2 + 8N − 8 = 4 n
+ mR
2R 2 ˜
+ 8N − 8 (2.1.5.12) where we have used the result of section 2.1.2 for the normal ordering. In particular,
the level matching condition (the second equality in (2.1.5.12)) implies that
˜
N − N = nm. (2.1.5.13) Thus, for zero winding and momentum number the spectrum coincides with the spectrum of the uncompactiﬁed string (see section 2.1.2). In the massless sector we have
again a graviton, antisymmetric tensor and dilaton which are obtained from the state
j αi 1 α−1 k
−˜ , i, j = 25. (2.1.5.14) The target space interpretation of the remaining excitations (containing creator(s) in
25th direction) is diﬀerent. The two states
αi 1 α251 k
− ˜− , α251 αi 1 k
− ˜− (2.1.5.15) are target space vectors. They correspond to gauge ﬁelds of a U (1) × U (1) gauge
symmetry. Finally, the state
α251 α251 k
− ˜− (2.1.5.16) 2. Toroidal Compactiﬁcation and Tduality 45 describes a target space scalar. The spectrum is supplemented by a KaluzaKlein
and winding tower of additional states as n, m run through the integer numbers. An
interesting question is wether some of these additional states are massless. For massless
states the mass shell condition (2.1.5.12) reads
2N − 2 + n
− Rm
2R 2 ˜
= 2N − 2 + n
+ Rm
2R 2 = 0. (2.1.5.17) These equations can be solved for nonvanishing n or m only at special values of R.
The most interesting case is23
R2 = 1
=α.
2 (2.1.5.18) One obtains the additional solutions listed in table 2.2.
n
1
1
1
1
2
2
0
0 m
1
1
1
1
0
0
2
2 N
1
1
0
0
0
0
0
0 ˜
N
0
0
1
1
0
0
0
0 Table 2.2: Each line in this table gives a conﬁguration of winding, momentum and
1
occupation numbers leading to massless states at R2 = 2 .
Each of the ﬁrst four states in table 2.2 contains one creator. This gives four
additional massless vectors (if the creator points into a noncompact direction) and four
massless scalars (if the creator points into the 25th direction). The latter four states
in table 2.2 correspond to massless scalars. Together with (2.1.5.15) and (2.1.5.16) we
have six vectors and nine scalars. The vectors combine into an SU (2) × SU (2) gauge
ﬁeld whereas the scalars form a (3, 3) representation. For the special value (2.1.5.18)
the gauge group U (1) × U (1) is enhanced to the nonabelian group SU (2) × SU (2).
The rank of the gauge group is not changed.
An immediate question is: what is so special about (2.1.5.18)? To answer this, we rewrite (2.1.5.12) in a suggestive way
˜
M 2 = 4N + 4N − 8 +
23 n2
+ 4m2 R2,
R2 (2.1.5.19) Later, in section 2.2.1, we will also discus the case R2 = 2, where less states become massless. 2. Toroidal Compactiﬁcation and Tduality 46 where we already aplied (2.1.5.13). We observe that the spectrum is invariant under
n ↔ m and R ↔ α
.
R (2.1.5.20) Recall that in the previous equations we have set α = 1/2. The symmetry (2.1.5.20)
is called Tduality. Winding and momentum numbers are interchanged and simultaneously the compactiﬁcation radius is inverted. If R takes the value (2.1.5.18), the
spectrum is invariant under interchanging winding with momentum. This radius is
called the selfdual radius. Because of the symmetry (2.1.5.20) we can restrict the
compactiﬁcations to radii equal or larger than (2.1.5.18). This is an important diﬀerence to the point particle discussed in the previous section. To make this diﬀerence
clearer let us take the compactiﬁcation radius to zero. All the KaluzaKlein momenta
diverge and only states with n = 0 survive. This is similar to the point particle case.
On the other hand, all winding states degenerate. In order to make sense out of
this situation one can apply the Tduality tranformation (2.1.5.20). But then R = 0
leads to the decompactiﬁcation limit and we are back at the 26 dimensional string.
Therefore, in string theory there are always traces of compact dimensions left.
Compactifying the string on a D dimensional torus, the above considerations lead
to a ZD symmetry in a straightforward way. However, combining the Tduality along
2
circles with basis redeﬁnitions of the torus lattice and integer shifts in the internal B
ﬁeld leads to an enhancement of the Tduality group to SO (D, D, Z).
2.1.5.3 Tduality in non trivial backgrounds In this section we will argue that the above described Tdualiy is also a symmetry for
nontrivial background conﬁgurations. We closely follow[392]. Our starting point is
the nonlinear sigma model (2.1.3.1). Compactiﬁcation of one target space dimension
is possible if the sigma model is invariant under constant shifts in this direction. For
the ﬁrst term in (2.1.3.1) this implies that the tangent to the compactiﬁed direction
is a Killing vector. The second term is invariant provided that the Lie derivative of
Bµν in the Killing direction is an exact twoform. For the last term to be invaraint
the Lie derivative of the dilaton Φ must vanish. We now choose coordinates such that
the isometry is represented by a translation in the d − 1 direction
X d−1 → X d−1 + c. (2.1.5.21) We call the other coordinates X i . The previously mentioned conditions on Bµν and
Φ imply that those ﬁelds are independent of X d−1 (up to gauge transformations).
The next step is to gauge the symmetry (2.1.5.21) and to “undo” this by constraining the gauge ﬁelds to be of pure gauge. The constraint is implemented with the help 2. Toroidal Compactiﬁcation and Tduality 47 of a Lagrange multiplier λ which ﬁnally will replace X d−1 in the Tdual model. We
introduce two dimensional gauge ﬁelds Aα changing under (2.1.5.21) as
Aα → Aα − ∂α c, (2.1.5.22) ∂α X d−1 → DαX d−1 ≡ ∂α X d−1 + Aα . (2.1.5.23) and replace Together with the above mentioned constraint (implemented by a Lagrange multiplier)
this amounts to adding to (2.1.3.1) (for simplicity we choose γαβ = ηαβ )24 a term
SA = − 1
4πα d2 σ (Gd−1,d−1 Aα Aα + 2Gd−1,ν Aα ∂α X ν
+2 αβ Bd−1,ν Aα ∂β X ν + 2λ αβ ∂ α Aβ . (2.1.5.24) Integrating over λ will result in the constraint of vanshing ﬁeld strength for the Aα
which in turn imposes
Aα = ∂α ϕ, (2.1.5.25) with ϕ being a worldsheet scalar. Shifting X d−1 by ϕ gives back the original sigma
model (2.1.3.1). Thus, adding (2.1.5.24) does not change anything. However, there
is a subtlety here. Compactifying the d − 1 direction means that we identify X d−1
with X d−1 + 2π (this time we put the compactiﬁcation radius into the target space
metric). In order to be able to absorb ϕ into X µ , ϕ should respect the same periodicity.
This can be ensured as follows. We continue the worldsheet to Euclidean signature
and study the sigma model for a torus worldsheet. Then we can assign two winding
numbers (corresponding to the two cycles of the torus) to the Lagrange multiplier
λ. Summing over these winding numbers (in a path integral approach) will impose
the required periodicity on the gauge ﬁelds Aα . Going through the details of this
prescription leads to the conclusion that the λ “direction” is compact λ ≡ λ + 2π .
Instead of integrating out λ (to check that we did not change the model) we can
integrate out Aα . (Since Aα is not a propagating ﬁeld this can be done by solving the
equations of motion. As well, one can integrate out Aα in a path integral, which is
Gaussian.) This procedure leads us to a dual model
1
4πα
1
−
4π S=− d2 σ √
˜
˜
˜
−γγ αβ Gµν ∂α X µ ∂β X ν + √
˜
d2 σ −γ ΦR(2). αβ ˜
˜
˜
Bµν ∂α X µ∂β X ν
(2.1.5.26) 24
For the Minkowskian worldsheet signature the i in front of the Bµν coupling in (2.1.3.1) is replaced
by one. 2. Toroidal Compactiﬁcation and Tduality 48 ˜
The set of dual coordinates is X µ = λ, X i , and the dual background ﬁelds are,
˜
Gd−1,d−1 = Bd−1,i
˜
, Gd−1,i =
,
Gd−1,d−1
Gd−1,d−1
1 Gi,d−1 Gd−1,j + Bi,d−1 Bd−1,j
˜
Gij = Gij −
,
Gd−1,d−1
Gi,d−1 Bd−1,j + Bi,d−1 Gd−1,j
Gd−1,i
˜
˜
, Bij = Bij +
.
Bd−1,i =
Gd−1,d−1
Gd−1,d−1 (2.1.5.27) (2.1.5.28) (2.1.5.29) To ﬁnd the dual expression for the dilaton is a bit more complicated. One can compute
˜
˜
Φ in a perturbative way. To this end, one requires that Φ is such that the conformal invariance conditions (2.1.3.53), (2.1.3.54), (2.1.3.55) are satisﬁed whenever the original
background satisﬁes them. This leads to the following formula for the dual dilaton
√
˜
e−2Φ −G = e−2Φ ˜
−G. (2.1.5.30) From a path integral perspective the dilaton transformation can be motivated as follows. The path integral measure for the X µ is covariant with respect to the metric
Gµν . In the dual model one would naturally use a measure which is covariant with re˜
spect to the dual metric Gµν . The change of the measure introduces a Jacobian which
leads to (2.1.5.30). To our knowledge this is a rather qualitative statement which is
diﬃcult to prove explicitly.
To make contact with the simple case discussed in the previous section we should
take Gd−1,d−1 = R2 /α , Gij = ηij , Bµν = 0 and Φ = const. Then the Tduality formulæ (especially (2.1.5.27)) imply that the compactiﬁcation radius is inverted. The
dilaton receives a constant shift and nothing else changes. This dilaton shift was not
visible in the discussion of the previous section. On the other hand, in the present
section we did neither see that Tduality interchanges winding with momentum nor
that there is an enhancement of gauge symmetry at the selfdual radius, because we
did not study the spectrum of the general string theory.
2.1.5.4 Tduality for superstrings In extending the discussion of section 2.1.5.2 to the superstring we will be sketchy
and omit the technical details. Most of the statements from section 2.1.5.2 can be
directly taken over to the superstring. In the Ramond sector, some new ingredients
occur. First, we consider the type II superstring. Instead of the 26th direction we
compactify the tenth direction. Combining the Tduality transformation (2.1.5.20)
with the mode expansions (2.1.5.9) and (2.1.5.10) one realizes that (2.1.5.20) can be 2. Toroidal Compactiﬁcation and Tduality 49 9
9
9
9
achieved by assigning X 9 = XL − XR , instead of X 9 = XL + XR, or equivalently
change (right movers) −→ − (right movers) (2.1.5.31) while keeping the original prescription of combining left with right movers. Carrying
this prescription over to the fermionic sector we observe that in the right moving
Ramond sector (see (2.1.2.58) for the deﬁnition)
(−)F −→ − (−)F . (2.1.5.32) This in turn implies that the Tduality transformation takes us from the type IIA
to type IIB string and vice versa. Hence, Tduality is not a symmetry in type II
superstrings but relates the type IIA string with type IIB. (This is true, whenever we
perform Tduality in an odd number of directions.) The type IIA string compactiﬁed
on a circle with radius R is equivalent to the type IIB string compactiﬁed on a circle
with radius α /R. This is consistent with the observation that the massless spectra of
circle compactiﬁed type IIA and type IIB theories are identical as depicted in table
2.3, where µ, ν = 0, . . . , 8. IIA
IIB NSNS
Gµν , Bµν , Φ, Gµ9, Bµ9
Gµν , Bµν , Φ, Gµ9, Bµ9 RR
Aµ , A9, Cµνρ , Cµν 9
Bµ9 , Φ , Cµνρ9 , Bµν Table 2.3: Massless type II ﬁelds in nine dimensions
In order to discuss compactiﬁcations of the heterotic string, it is useful to employ
a formulation where the additional 32 left moving fermions are bosonized into 16 left
moving bosonic degrees of freedom. We will not carry out this construction here.
It can be found in the books listed in section 6.1.1. The result which is of interest
in the current context is that those 16 left moving bosons are compactiﬁed on an
even selfdual lattice25. That is, that even without further compactiﬁcations from
ten to less dimensions the heterotic string contains already a leftright asymmetric
compactiﬁcation. The theory does not depend on changing the basis of the ‘internal’
lattice. Compactifying the tenth dimension one observes another new feature which
is present in the heterotic string. In the previously discussed cases, there was one
modulus in circle compactﬁcations, viz. the radius of the circle. For the heterotic
string we have 16 more moduli. These are called Wilson lines. They arise from the
25 There exist exactly two such lattices Γ8 × Γ8 and Γ16 giving rise to the E8 × E8 and the SO(32)
string, respectively. 2. Orbifold ﬁxed planes 50 possibility that the nonabelian gauge ﬁelds can take constant vacuum expectation
values (vev) in the Cartan subalgebra of the gauge group. The fact that (at least) one
of the ten directions has been compactiﬁed is important here. Otherwise, a constant
vev could be gauged away. To see this explicitly, let us assume that the gauge ﬁeld
vev is (proportional to a generator in the Cartan subalgebra)
A9 = x9 Θ
x9 Θ
Θ
= e − R ∂9 e R ,
R (2.1.5.33) where the second part of the equation shows that a constant vev is a pure gauge
conﬁguration. However, in the compact case we have identiﬁed x9 with x9 + 2πR and
hence only gauge transformations which are periodic under this shift are allowed. This
implies that the Wilson line (2.1.5.33) can be gauged away only if Θ is an integer. From
this discussion it follows that generically the gauge group is broken to U (1)16 in the
compactiﬁcation process. In addition there are the (abelian) KaluzaKlein gauge ﬁelds
G9µ and B9µ corresponding to a U (1) × U (1) gauge symmetry. Thus, generically there
is a U (1)18 gauge symmetry in the circle compactiﬁed heterotic string. Depending
on the moduli (Wilson lines and compactiﬁcation radius) there are special points of
stringy gauge group enhancement.
It can be proven that the E8 × E8 heterotic string and the SO(32) string are continuously connected in moduli space once they have been compactiﬁed to nine
dimensions. This can be shown by observing that for a certain conﬁguration of Wilson
lines (where the gauge group is broken to SO(16) × SO(16) in either theory) T duality maps the two compactiﬁcations on each other. (For details see e.g. Polchinski’s
book[371].) All other compactiﬁcations can be reached by continuously changing the
moduli. Including the original ten dimensional theories as decompactiﬁcation limits
we see that the two diﬀerent heterotic strings belong to the same set of theories sitting
at diﬀerent corners in moduli space. For completeness we should mention that for
compactiﬁcations of the heterotic string on a D dimensional torus one ﬁnds the Tduality group SO (16 + D, D, Z). 2.2 Orbifold ﬁxed planes In the previous sections we have studied the theory of a one dimensional extended
object – the string. One of the striking features of this theory is that it automatically
also describes objects which are extended along more than one space direction. As the
simplest example we will study now the orbifold ﬁxed planes. Here, one compactiﬁes
the string on a torus whose lattice has a discrete symmetry, and gauges this symmetry26. Thus, the compact manifold is a D dimensional torus divided by some discrete
26 With gauging of a discrete symmetry we mean that only invariant states are kept. 2. The bosonic string on an orbicircle 51
¨¡¨§
§§¡
¨
§¨¡
¡§¨§¨ ≡ ¢¡¢ ¡
¢ ¢¡
¡ ¢ ¢ ¤¡¤£
££¡
¤
£¤¡
¡£¤£¤ ¦¡
¥
¥¡
¡¥¥¦¦¦¥
¦¡
¥¦¡ Figure 2.4: The interval as an orbicircle. The ﬁxed points (black dots) form the ends
of the interval.
group. (We will consider Z2 as such a group. It leaves an arbitrary lattice invariant.)
There are some points or —when combined with the other directions— planes which
are invariant under the discrete group. These are the orbifold ﬁxed planes. They
present singularities in the compact part of the space time. String theory gives a
physical meaning to orbifold ﬁxed planes. We will see that certain string excitations
(particles or gauge ﬁelds from the target space perspective) are conﬁned to be located
at the orbifold ﬁxed planes. By looking at an example where the orbifold can be
reached as a singular limit of a smooth manifold we will see that for string theory the
singular nature of this limit is not “visible”. Instead of discussing the general setups
for orbifold compactiﬁcations we will present two examples: the bosonic string on an
orbicircle and the type IIB string on T 4 /Z2. We hope that this will demonstrate the
general idea with a minimal amount of technical complications. For more details (and
also orbifold compactiﬁcations of the heterotic string) we recommend the review[354]. 2.2.1 The bosonic string on an orbicircle Let us start by describing the target space geometry. We compactify the 25th dimension on a circle like in section 2.1.5.2. In addition, we identify opposite points on this
circle. If we choose the “fundamental domain” to be −πR < x25 < πR this is done
by the Z2 identiﬁcation: x25 ≡ −x25 . The resulting target space is an interval in the 25th direction as depicted in ﬁgure 2.4. Taking into account the uncompactiﬁed
dimensions, the end points of the interval (the ﬁxed points of the Z2 ) correspond to
planes with 24 spatial directions. Therefore, we call them orbifold24planes.
We proceed by constructing the untwisted spectrum. The term untwisted (in
contrast to twisted) will become clear later. It means that we construct the spectrum 2. The bosonic string on an orbicircle
State
αi 1 0
−
α251 0
− Z2
+
− 52 24 + 1 dim. rep.
1 vector
1 scalar Table 2.4: Untwisted right moving states
which is invariant under the orbifold projection x25 → −x25 . Since in the bosonic
string the right moving sector is identical to the left moving one, we ﬁrst write down
the right moving states only. The result is collected in table 2.4 (i = 2, . . . , 24; the
zeroth and ﬁrst direction are eliminated by the lightcone gauge).
Now we need to combine left with right movers such that the resulting state is
invariant under the Z2 . This is the case for the product of the vector with the vector
and the scalar with the scalar. Thus, we obtain the metric Gij , the antisymmetric
tensor Bij and the dilaton Φ. The additional U (1) vectors Gi25 and Bi25 are projected
out in contrast to section 2.1.5.2. The combination of the scalar from the left moving
sector with the scalar from the right moving sector yields a target space scalar G25 25 .
Since the groundstate is Z2 invariant, the tachyon will survive the projection. If we are
at the selfdual radius, there might be additional massless states (without imposing Z2
invariance these are listed in table 2.2). The action of the Z2 takes winding number
to minus winding number and momentum number to minus momentum number as
can be seen from the mode expansion (2.1.5.9), (2.1.5.10). This means that we can
keep only invariant superpositions of states. From the ﬁrst four entries in table 2.2
we obtain two additional massless vectors. These arise as follows. We add the ﬁrst
state to the second state of the listing and act with αi 1 , or we add the third to the
−
˜
fourth state and act with αi . We can also subtract the second from the ﬁrst state
α251 ,
− −1 and act with
or we subtract the fourth from the third state and act with α251 .
˜−
This gives two massless scalars. Adding the ﬁfth to the sixth entry and the seventh to
the eighth, we obtain two more scalars at the selfdual radius. This looks very unusual.
Since we do not have any U (1) gauge ﬁelds away from the selfdual radius, now also the
rank of the gauge group is enhanced at the selfdual radius. There are also additional
tachyons at the selfdual radius. These are the two states which are obtained by adding
the n = 0, m = 1 vacuum to the n = 0, m = −1 vacuum. The second state is the
same with m and n interchanged. These two additional tachyons have mass squared
M 2 = −6, as can be easily computed from (2.1.5.19).
Now, we come to a new feature which is unique to string theory. There are additional twisted sector states. These states are periodic under shifting σ by π only up
to a (nontrivial) Z2 transformation. In our case this implies for the string that its
center of mass position has to be located at a ﬁxed plane and that the integer Fourier 2. The bosonic string on an orbicircle 53 modes are replaced by halfinteger ones in the 25th direction. In the twisted sector we
need to compute the groundstate energy. This can be done by ﬁrst modifying equation
(2.1.2.46) in a straightforward way
atwisted = − ∞ 1
23
n−
2 n=1
2 ∞ r. (2.2.1.1) r= 1
2 Regularizing this expression according to the prescription (2.1.2.47) gives a = 15
16 . This implies that the groundstate is tachyonic and also that there is no massless state
coming from this twisted ground state. There is one more tachyonic state at the ﬁrst
˜−
level in the twisted sector. This is obtained by acting with α251 α251 . Collecting the
−
2 2 7
results, we obtain one tachyon with M 2 = − 15 and one with M 2 = − 2 at each ﬁxed
2
plane. Altogether, there are four tachyons (and states with positive mass squared) located at the ﬁxed planes.
The singular nature of the ﬁxed points does not raise any problem in string theory.
It introduces twisted sector states which result in additional particles which are located
at the orbifold24planes in target space.
It is interesting to note that the orbifold at the selfdual radius is equivalent to the
toroidally compactiﬁed bosonic string at twice the selfdual radius.27 For a detailed
disussion of this equivalence we refer to Polchinski’s book[371]. Here, we will only
check that the light (tachyonic and massless) spectra coincide. Obviously, the gauge
groups U (1) × U (1) are the same. For the bosonic string on the circle (with R2 = 2) these are the oﬀdiagonal metric and B ﬁeld components Gi25, Bi25 , whereas for the
1
orbicircle compactiﬁcation at R2 = 2 these come from states with nontrivial winding
and momenta as discussed above. It remains to identify the four additional massless
scalars and the two tachyons (found in the nontrivial windingmomenta sector) of the orbicircle compactiﬁcation (at selfdual radius) and the four additional tachyons from
the twisted sector in the circle compactiﬁcation. Here, the special choice R2 = 2 for
the circle compactiﬁcation comes into the game. With (2.1.5.19) evaluated at R2 = 2
we ﬁnd exactly these missing states. At ﬁrst, there are four massless scalars: the vacua
with m = 0 and n = ±4, or m = ±1 and n = 0. The two tachyons with M 2 = −6 are
obtained from the two vacua with m = 0 and n = ±2. The two twisted sector tachyons with M 2 = − 15 correspond to the two vacua with m = 0 and n = ±1. The other two
2
7
twisted sector tachyons with M 2 = − 2 can be identiﬁed in the circle compactiﬁcation
as the vacua with m = 0 and n = ±3. The equivalence of the S 1 /Z2 compactiﬁcation at the selfdual radius and the S 1
compactiﬁcation at twice the selfdual radius shows that the moduli spaces of both
27 Alternatively, we could use the Tdual version at half the selfdual radius. 2. Type IIB on T 4 /Z2 54
x9 x8 Figure 2.5: The orbifold T 2/Z2
compactiﬁcations are connected at this point. This feature has a stringy origin. From
the target space perspective this is quite surprising. A ﬁeld theory on 24 + 1 dimensional Minkowski space times an interval with certain ﬁelds constrained to live at the
endpoints of the interval is smoothly connected to a ﬁeld theory on 24 + 1 dimensional
Minkowski space times a circle with all ﬁelds living in the whole space. However, due
to the tachyons both vacua are unstable. In the next section we will see that similar
things happen for the superstring which does not have tachyons in its spectrum. 2.2.2 Type IIB on T 4/Z2 Again, we start by describing the target space geometry. We compactify the six, seven,
eight and nine direction on a four dimensional torus. We view this four dimensional
torus as the product of two twodimensional tori. The coordinates are labeled such
that the sixth and seventh direction form one T 2 and the eighth and ninth a second
T 2. Let us focus on this second T 2 with the understanding that the same applies to the
ﬁrst T 2 . In ﬁgure 2.5 this is depicted by drawing a lattice in the eightnine plane. The
fundamental cell is the parallelogram with edges drawn with stronger lines. The lattice
vectors are the lower and the left edge of the fundamental cell. A two dimensional
torus is obtained by gluing together the opposite edges of the fundamental cell. Shifts
by lattice vectors connect identiﬁed points.
“Dividing” the T 4 by Z2 means that in addition we identify points via the prescription
x6 , x7, x8, x9 → −x6 , −x7, −x8, −x9 . (2.2.2.1) This Z2 action leaves the four points indicated by black dots in ﬁgure 2.5 times the 2. Type IIB on T 4 /Z2
Sector
NS: 55
Z2
+ 5 + 1 dim. rep.
1 vector − State
i
ψ− 1 0 4 scalars e
+
− 2 (antichiral) spinors
2 (chiral) spinors 2 i = 2, . . . , 5
6,7,8,9
ψ− 1 0
2 R: s1 s2 s3 s4
s1 = s2 , s3 = −s4
s1 = −s2 , s3 = s4 i
(s3 +s4 )
2 Table 2.5: Untwisted right moving states
four points in the ﬁrst torus invariant. Thus, we obtain sixteen orbifold ﬁveplanes. At
ﬁrst, we construct the untwisted spectrum. Since in the type IIB case the left moving
sector is identical to the right moving one, we ﬁrst write down the right moving states,
only. The result is collected in table 2.5. We choose the GSO projection such that
—in the notation of (2.1.2.56)— states with an odd number of minus signs survive.
The projection (2.2.2.1) can be viewed as rotations by 180◦ in the eightnine plane
and simultaneously in the sixseven plane. This is useful for the identiﬁcation of the
behavior of the Rsector under Z2 transformations. We consider only states which lead
to massless particles when combined with the left movers.
In the NSNS sector, we can combine the left moving vector with the right moving
one leading to the six dimensional graviton Gij , the antisymmetric tensor Bij and the
dilaton Φ. Further, we can combine scalars from the left moving sector with scalars
from the right moving one. This gives sixteen massless scalars corresponding to Gab
and Bab where the indices a, b are internal, i.e. a, b = 6, . . . , 9. The target space vectors
Gia and Bia are projected out. In the RR sector, we can combine the chiral spinor with
the chiral one, and the antichiral with the antichiral one. This leads to 32 massless
(onshell) degrees of freedom in the RR sector. Tensoring a chiral spinor with a chiral
spinor gives a selfdual twoform potential (3 on shell components) and a scalar. The
tensor product of two antichiral spinors yields an antiselfdual twoform potential and
a scalar. We can perform four of those combinations, each. With the notation of table
2.1 the RR states can be identiﬁed as follows:
∗
∗
• Cijkl (or Cabcd ) gives 4
4 = 1 degree of freedom (one scalar), ∗
• Cabij gives 3 antiselfdual twoforms and 3 selfdual twoforms (18 degrees of
freedom), • Bij gives a twoform (6 degrees of freedom),
• Bab gives six scalars, 2. Type IIB on T 4 /Z2 56 • Φ gives a scalar .
All other ﬁelds from the RR sector are projected out. Fermionic degrees of freedom
are obtained from the NSR and RNS sector. Combing the vector with the antichiral
spinors gives four times28a (2, 2) ⊗ (2, 1) = (3, 2) ⊕ (1, 2) representation of the six
dimensional little group SO (4) = SU (2) × SU (2).29 Therefore, this tensor product
provides us with four chiral gravitini and four chiral fermions.
Combining the NS sector scalars with the chiral R sector spinors gives 16 chiral spinors. From the existence of the four chiral gravitini we can guess that the resulting
low energy eﬀective ﬁeld theory has N = 4 chiral supersymmetry in six dimensions.
(For a collection of supersymmetries in various dimension see[399].)
Before checking that also the rest of the massless states ﬁt into supersymmetric
multiplets we should construct the twisted sector states. The construction does not
depend on the location of the ﬁxed plane. Therefore, we restrict the construction
to one plane and multiply the result by 16. In the twisted sector, the NS fermions
with an index corresponding to a compact dimension are integer modded whereas the
R sector fermions are half integer modded. Now, there are NS sector zero modes
forming a four dimensional Cliﬀord algebra. The twisted NS ground state is twofold
degenerate after imposing the GSO projection. (We modify the notation of (2.1.2.56)
in a straightforward way. Since we have only two creators and two anihilators, the
twisted NS groundstate has two entries. Performing the GSO projection means that
we keep those states with an odd number of minus signs.) In the twisted R sector
we do not have zero modes in the compact direction. This lifts some of the vacuum
degeneracy as compared to the untwisted sector. The twisted R ground state is labeled
only by the ﬁrst two entries. Again, we keep only states with an odd number of minus
signs. In order to deduce the masses of the states in the twisted NS sector, we observe
from (2.1.2.46) (and its regularisation) that replacing four integer moded bosons by
4
5
4
half integer moded ones changes the normal ordering constant by − 24 − 16 = − 12 . Changing the modding of four worldsheet fermions from halfinteger to integer gives
4
4
1
another shift of − 16 + 24 = − 12 . Thus, we arrive at
atwisted = auntwisted −
NS
NS 1
= 0.
2 (2.2.2.2) The twisted NS sector groundstate is massless. The R sector groundstate is always
massless, since fermions have the same modding as bosons. The analogon of table
2.5 for the twisted sector is table 2.6. Since all the twisted sector groundstates are
28 One factor of two arises because the NSR and RNS sector yield such a tensor product, each. The
second factor of two is due to the two antichiral spinors in table 2.5.
29
A vector is in the (2, 2), an antichiral spinor in the (2, 1), a chiral spinor in the (1, 2), a selfdual
twoform in the (3, 1) and an antiselfdual twoform in the (1, 3). 2. Comparison with type IIB on K 3
Sector
NS:
R: State
s3 s4
s3 = −s4
s1 s2
s1 = −s2 , Z2 57
5 + 1 dim. rep. iπ
(s3 +s4 )
2 e
+ 2 scalars + 1 (chiral) spinors Table 2.6: Twisted right moving states
invariant under the Z2 we can form all possible leftright tensor products. Multiplying
with 16 (the number of ﬁxed planes) we obtain 64 scalars from the NSNS sector. The
RR sector leads to 16 antiselfdual twoforms and 16 scalars. The RNS and NSR sector
give rise to 64 chiral spinors.
After we have obtained the full massless spectrum of the type IIB string on T 4 /Z2
we can ﬁt it into supermultiplets of N = 4 chiral supergravity in six dimensions. The
possible supermultiplets are the gravity multiplet and the tensor multiplet. The gravity
multiplet contains the graviton and four chiral gravitini from the untwisted sector. In
addition, ﬁve selfdual twoforms are in the gravity multiplet. A tensor multiplet is
made out of an antiselfdual twoform, ﬁve scalars and four chiral fermions. The ﬁve
∗
selfdual twoforms in the gravity multiplet we take from Bij , Bij and Cijab . After
ﬁlling the gravity multiplet, we are left with 21 antiselfdual twoforms, 105 scalars
and 84 chiral fermions. Thus, the remaining degrees of freedom ﬁt into 21 tensor multiplets.
To summarize, the massless spectrum of the type IIB string on T 4 /Z2 consists of
one gravity multiplet and 21 tensor multiplets of D = 6 chiral N = 4 supersymmetry.
Some of the degrees of freedom are conﬁned to live on the orbifold5planes which
ﬁll the 5 + 1 dimensional noncompact space but are located in the four dimensional
compact space. In the next section we will argue, that this setup is smoothly connected
to compactiﬁcations without orbifold5planes. 2.2.3 Comparison with type IIB on K 3 In the previous section we compactiﬁed the type IIB string on T 4 /Z2. Among others,
we obtained four chiral gravitini. If we compactiﬁed on T 4 instead, the two ten dimensional gravitini would give rise to four nonchiral gravitini in six dimensions. Thus,
our orbifolding removes half of the supersymmetries. This is due to the fact, that
the T 4 /Z2 manifold belongs to a larger class of manifolds which are called CalabiYau
nfolds. Here, n denotes the number of complex dimensions, i.e. n = 2 in our case.
The CalabiYau twofolds are all connected by smooth deformations and commonly
denoted by K 3. One important feature of CalabiYau nfolds is that they possess 2. Comparison with type IIB on K 3 58 SU (n) holonomy. This means that (for K3) going around closed (noncontractable)
curves induces an SU (2) transformation. In a toroidal compactiﬁcation we split the
ten dimensional spinor into a couple of lower dimensional spinors. The possible values
of the internal spinor indices count the number of resulting lower dimensional spinors.
In a torus compactiﬁcation, each value of the internal indices gives rise to a massless
spinor. This is because the internal homogenous Dirac equation has always a solution
– any constant spinor. If instead of a torus with trivial holonomy we compactify on
K 3 with SU (2) holonomy, only spinors which do not transform under the holonomy
group give rise to massless six dimensional spinors. This removes half of the internal
components and thus breaks half of the supersymmetry. Indeed, all K 3 compactiﬁcations yield the same massless spectra. This is a consequence of the fact that the
number of zeromodes (of Laplace and Dirac operators) does not change as we move
from one K 3 to another one. The number of zero modes of the Laplace operators30
are usualy listed in Hodge diamonds. The Hodge diamond for K 3 is
1
0
1 0
1. 20
0 (2.2.3.1) 0
1 In the following we explain (roughly) how to read (2.2.3.1). The K 3 is a complex
manifold. Therefore, we can choose complex coordinates (and so we do). Then a
tensor can have a couple of holomorphic indices and a couple of antiholomorphic
indices. In other words, there are (p, q ) forms on K 3, where p corresponds to the
number of holomorphic indices and q to the number of antiholomorphic ones. Since
the complex dimension of K 3 is two, the values of p, q can be zero, one or two. We
denote the number of zero modes of a (p, q ) form with h(p,q). These (Hodge) numbers
are arranged into a Hodge diamond as follows31
h0,0
h1,0
h2,0 h0,1
h1,1 h2,1 h0,2 . (2.2.3.2) h1,2
h2,2 From (2.2.3.1) we can deduce that an object which represents a zero or a four form
in the internal space, has one zero mode. Such an object gives rise to one massless
30 There are several diﬀerent Laplace operators whose form depends on the tensor structure of the
object they act on.
31
The symmetry of (2.2.3.1) is not accidental. The vertical symmetry is related to Hodge duality
and the horizontal one to interchanging holomorphic with antiholomorphic coordinates. 2. Comparison with type IIB on K 3 59 sixdimensional ﬁeld. A p + q = 2 form possesses 22 zero modes, thus leading to 22
massless ﬁelds in six dimensions. In order to write down the massless spectrum of
the K 3 compactiﬁed type IIB string we need to know another feature of the family of
K 3 manifolds. All CalabiYau manifolds (and in particular K 3) are Ricci ﬂat. This
means that the Ricci tensor vanishes and hence we do not need any nontrivial background conﬁguration in order to satisfy the conformal invariance conditions derived in
section 2.1.3. This remains true under certain deformations of the metric of K 3. The
space of such nontrivial (not related to coordinate changes) metric deformations is 58
dimensional for the family of K 3s.
Now, we are ready to derive the bosonic massless spectrum of the K 3 compactiﬁed
type IIB string. At ﬁrst, we collect all zero forms of K 3. From the NSNS sector these
∗
are Gij ,Bij ,Φ, and from the RR sector Bij , Φ , Cijkl . Since h(0,0) = 1 these appear
once in the lower dimensional spectrum. The Gab are not diﬀerential forms on K 3 but
metric deformations. They result in 58 massless scalars. Since h(p,q) = 0 for p + q odd,
the KaluzaKlein ﬁelds Gia and Bia do not give rise to massless six dimensional ﬁelds.
∗
It remains to count the twoforms on K 3. (The four form Cabcd we have already
∗
∗
counted as Cijkl because of selfduality.) The twoforms are Bab , Bab and Cijab . Bab
∗
and Bab lead to 44 scalars in six dimensions. The 22 zeromodes of Cijab can be decomposed into three selfdual and 19 antiselfdual twoforms in six dimensions[38].
Taking into account that the SU (2) holonomy breaks half of the supersymmetry (as
compared to T 4 compactiﬁcations) and that the fermionic zero modes are all of the
same chirality, we obtain the same massless spectrum as in the T 4 /Z2 case.
Indeed, T 4 /Z2 corresponds to a limit in the space of K 3 manifolds where the K 3
degenerates. As long as one considers K 3s very close to that point one obtains the
same massless spectrum. In string theory even the limit to the point where the K 3
degenerates is well deﬁned.
Let us see what happens when we repeat the K 3 analysis for the orbifold T 4/Z2 . We
will focus only on the bosonic spectrum. First, we need to know the Hodge diamond for
T 4/Z2 . This can be easily “computed” without much knowledge of algebraic geometry.
On T 4 we obtain the Hodge numbers just by counting independent components of the
corresponding diﬀerential forms,
hp,q = 2
p 2
q . (2.2.3.3) The Z2 action is taken into account by removing forms which are odd under the Z2 . 2. Comparison with type IIB on K 3 60 Thus, the Hodge diamond for T 4 /Z2 is
1
0
1 0
1. 4
0 (2.2.3.4) 0
1 ∗
In six dimensions we obtain two twoforms Bij and Bij and three scalars Φ, Φ , Cijkl .
The internal metric components Gab yield ten scalars. (Note, that constant rescal ings of the coordinates would change the range of those coordinates, and hence are
nontrivial deformations equivalent to a constant change of the corresponding metric
components.) From the metric deformations we obtain 48 less scalars than in the K 3
compactiﬁcation. It remains to take into account the twoforms on T 4 /Z2: Bab , Bab
∗
and Cijab . We obtain 12 massless scalars from Bab and Bab , together. On the K 3 there
∗
were 32 more massless scalars coming from this sector. The Cijab combine into three
selfdual and three antiselfdual twoforms. In the K 3 compactiﬁcation we obtained 16 more antiselfdual twoforms.
The T 4 /Z2 spectrum we computed here, would correspond to the one which we
had obtained in a ﬁeld theory compactiﬁcation. It has 80 massless scalars and 16
antiselfdual twoforms less than the K 3 compactiﬁed theory. In ﬁeld theory, the spectrum jumps when we take the singular orbifold limit in the family of K 3s. From the
above construction it is obvious that we counted only untwisted states from a string
perspective. Indeed, the missing 80 scalars and 16 antiselfdual twoforms are exactly
what we obtained from the twisted sector in the previous section. In string theory the
spectrum of the compactiﬁed theory does not feel the singular nature of the orbifold
limit. All that happens is that some part of the spectrum is localized to the orbifold
ﬁxed planes. This localization appears in internal space and is not visible at experiments which cannot resolve the distances of the size of the compact manifold. The
energy scale of such experiments depends on the type of interactions ﬁelds propagating
into the compact directions carry. For purely gravitational interactions it needs to be
much higher than e.g. for electromagnetic interactions. We will come back to this
later.
To summarize we recall that string theory can be compactiﬁed on singular manifolds. The moduli spaces of such compactiﬁcations can be connected to compactiﬁcations on smooth manifolds. There are massless states which are localized at singularities of the compact manifold. These are the twisted sector states. They are of truly
stringy origin. 2. Dbranes 2.3 61 Dbranes In this section we will present another kind of extended objects resulting from string
theory — the Dbranes. They are diﬀerent from the previously studied orbifold planes.
Dbranes can exist also in uncompactiﬁed theories. They are dynamical objects, i.e.
they interact with each other and can move independent of the size of some compact
space. (The orbifold planes could move only if we changed the size or shape of the
compact manifold.) When we discussed the fundamental string we did not consider
the possibility of open strings. We will catch up on that in the following. 2.3.1 Open strings 2.3.1.1 Boundary conditions We recall the action of the superstring
S= 1
2πα dσ + dσ − iµ
iµ
∂− X µ∂+ Xµ + ψ+ ∂− ψ+µ + ψ− ∂+ ψ−µ .
2
2 (2.3.1.1) Now, we view the values σ = 0, π as true boundaries of the string worldsheet. Varying
(2.3.1.1) with respect to X µ gives apart from the equations of motion (which are
identical to closed strings) boundary terms which should vanish separately,
δX µ∂σ Xµ π=0 = 0.
σ (2.3.1.2) For the closed string we have solved this equation by relating the values at σ = 0 with
the ones at σ = π . This procedure was local because we took the string to join to a
closed string at σ = π . Now, we proceed diﬀerently by not correlating the two ends of
the string, i.e.
δX µ ∂σ Xµ σ=0 = δX µ∂σ Xµ σ=π = 0. (2.3.1.3) Let us focus on the boundary at σ = 0. We have two choices to satisfy the boundary
condition. If —for i = 0, . . . , p— we allow for free varying ends (δX i arbitrary at the
boundary) we obtain Neumann boundary conditions at those ends32
∂σ X i = 0 , i = 0, . . . , p. (2.3.1.4) For the remaining d − p − 1 coordinates X a we choose to freeze the end of the string — the variation vanishes at the boundary. Hence, in those directions the end of the
string is conﬁned to some constant position. The resulting boundary conditions are
Dirichlet conditions (ca is a constant vector),
X a = ca , a = p + 1, . . . , d.
32 (2.3.1.5) Here, we use i to label Neumann directions. After ﬁxing the light cone gauge we take i = 2, . . . , p. 2. Open strings 62 The end of the open string deﬁnes a surface which extends along p + 1 dimensions and
is located in d − p − 1 dimensions. This object is called Dbrane, where the “D” refers to the Dirichlet boundary condition specifying its position. If we choose identical
boundary conditions for the other end of the open string (at σ = π ), we describe
an open string starting and ending on the same Dbrane. For diﬀerent boundary
conditions the open string stretches between two diﬀerent Dbranes. The Neumann
conditions imply that no momentum can ﬂow out of the ends of the open string. In the
Dirichlet directions momentum can leave the string through its end — it is absorbed
by the Dbrane. The target space Lorentz group is broken to SO(p, 1).
Varying the action with respect to the worldsheet fermions results in the same
equations of motions as in the closed string case and in the boundary conditions
µ
µ
−ψ+µ δψ+ + ψ−µ δψ− π
σ =0 = 0. (2.3.1.6) In the closed string case we have solved this by assigning either periodic or antiperiodic boundary conditions to the worldsheet fermions. Since now the ends of the
string are separated in the target space this would imply some nonlocality. Therefore,
we impose the boundary conditions (2.3.1.6) at each end separately
µ
µ
−ψ+µ δψ+ + ψ−µ δψ− σ =0 µ
µ
= −ψ+µ δψ+ + ψ−µ δψ− σ =π = 0. (2.3.1.7) Let us focus again on the boundary at σ = 0. We can solve (2.3.1.7) by one of the
options
µ
µ
ψ+ = ±ψ− at σ = 0. (2.3.1.8) However, there is a correlation with the bosonic boundary conditions via worldsheet
supersymmetry. To be speciﬁc, we choose the plus sign for Neumann conditions
i
i
ψ+ = ψ− at σ = 0. (2.3.1.9) The supersymmetry transformations (in particular (2.1.1.22) and (2.1.1.23)) should
not change this boundary condition. Since ∂τ X i is not speciﬁed by the boundary
conditions this yields
+ =− − at σ = 0, (2.3.1.10) which implies that due to the boundary (at least) half of the worldsheet supersymmetry
is broken. (If we had started with (1, 0) worldsheet supersymmetry —as we did for the
heterotic string— the boundary would break worldsheet supersymmetry completely.)
In order to ensure that not all of the supersymmetry is broken we have to choose 2. Open strings 63 the opposite (compared to (2.3.1.9)) boundary conditions for worldsheet fermions in
Dirichlet directions
a
a
ψ+ = −ψ− at σ = 0. (2.3.1.11) We could also interchange the fermionic boundary conditions in Dirichlet and Neumann directions. Then another worldsheet supersymmetry would survive. There is
no physical diﬀerence between the two choices. Nevertheless it is important, that we
take the boundary conditions in the Neumann directions to be “opposite” to the ones
in Dirichlet directions. One may also check that the open string action is invariant
under the worldsheet supersymmetries (2.1.1.20), (2.1.1.22) and (2.1.1.23) provided
that the worldsheet fermions satisfy the boundary conditions (2.3.1.9), (2.3.1.11) and
(2.3.1.10) is fulﬁlled. (Partial integration introduces boundary integrals which vanish
if these additional constraints hold.) Recall also that the functional form of the worldsheet supersymmetry parameter is restricted by the chirality conditions (2.1.1.28).
In the following we are going to discuss the boundary conditions at the other end
of the open string at σ = π . Going back to the closed string we deduce from (2.1.1.22)
that for antiperiodic supersymmetry parameter + the fermions are antiperiodic for
periodic bosons and vice versa. This means that antiperiodic + belongs to the
NS sector and periodic ones to the R sector. From the discussion of the boundary
conditions at σ = 0 we infer that the supersymmetry parameter has to satisfy one of
the following conditions,
+ =± − at σ = π. (2.3.1.12) In order to relate this to something like periodicity or antiperiodicity we perform
the so called doubling trick. This means that we deﬁne a function ε on the interval
0 ≤ σ < 2π . This is done in the following way (we indicate only the σ dependence),
ε= (σ )
, 0≤σ<π
.
± − (2π − σ ) , π ≤ σ < 2π
+ (2.3.1.13) The sign in the second line of (2.3.1.13) is correlated to the sign in (2.3.1.12) by the
requirement of continuity at σ = π . Hence, ε is (anti)periodic for the lower (upper)
sign in (2.3.1.12) (taking into account the sign in (2.3.1.10)).
Now, let us perform this doubling trick also on the worldsheet bosons and fermions.
For the bosons it is useful to rewrite the boundary conditions. Dirichlet boundary
conditions mean that
∂+ X a = −∂− X a at σ + − σ − = 0. (2.3.1.14) 2. Open strings 64 Neumann conditions can be written as
∂+ X i = ∂− X i at σ + − σ − = 0. (2.3.1.15) The next step is to specify the boundary conditions at σ = π . After having done this,
one can perform the doubling trick, i.e. deﬁne a boson ∂X µ on the interval 0 ≤ σ < 2π analogously to the deﬁnition of ε in (2.3.1.13) where ∂± X take the role of ± . As the
reader can easily verify, the outcome is that ∂X µ is periodic whenever we have chosen
the same type of boundary conditions at the two ends of the string in the xµ direction.
The corresponding open string sectors are called DD (NN) according to the choice of
Dirichlet (Neumann) boundary conditions at the two ends. For an opposite choice of
boundary conditions (ND or DN strings) ∂X µ will turn out to be antiperiodic. In
analogy to the closed string we call the sector with periodic ε R sector and the one
with antiperiodic ε NS sector. For DD or NN strings this implies that in the NS
sector we take the boundary conditions at σ = π to be
i
i
a
a
ψ+ = −ψ− , ψ+ = ψ− at σ = π. (2.3.1.16) Deﬁning a “doubled” worldsheet fermion Ψµ in analogy to ε (where the role of ± is
µ
taken over by the ψ± ) we ﬁnd that for DD or NN strings Ψ is antiperiodic. In the R
sector we take the boundary conditions a
a
i
i
ψ+ = ψ− , ψ+ = −ψ− at σ = π (2.3.1.17) and obtain periodic boundary conditions. In the above we have used that, for example,
in the Rsector periodicity of ε implies that the boundary conditions of ± at σ = π
are identical to the ones at σ = 0. Plugging this back into the supersymmetry transformations (2.1.1.22) and (2.1.1.23) evaluated at σ = π and taking into account the
boundary conditions for the bosons, we obtain the boundary conditions of the worldsheet fermions at σ = π . This in turn determines the periodicity of Ψµ . Performing
the same procedure for ND or DN boundary conditions, one ﬁnds that Ψµ is periodic
in the NS sector and antiperiodic in the R sector whenever xµ is a direction with ND
or DN boundary conditions. The ND or DN directions are somewhat similar to the
twisted sectors we met when discussing Z2 orbifolds.
At ﬁrst, we will consider only the case of a single Dbrane. This means that we can
have only DD or NN boundary conditions depending on whether we are looking at a
direction transverse or longitudinal to the Dbrane. Then ∂X µ will always be periodic
and Ψµ (anti)periodic in the (NS) R sector. 2. Open strings
2.3.1.2 65 Quantization of the open string ending on a single Dbrane The quantization of the open string is very similar to the closed superstring. In
the following we will point out the diﬀerences. At ﬁrst, we solve the equations of
motion again by taking a superposition of leftmoving and rightmoving ﬁelds. For
the bosons, these are given in (2.1.2.3) and (2.1.2.4). The boundary conditions relate
±
±
this two solutions. (In addition, we need to replace e−2inσ → e−inσ .)33 In Neumann
directions, they impose
αi = α i .
˜n
n (2.3.1.18) For Dirichlet directions, we obtain a similar relation and constraints on the zero modes,
xa = ca , pa = 0 , αa = −αa .
˜n
n (2.3.1.19) The general solutions for the bosonic directions read
X i = xi + pi τ + i
n=0 X a = ca − n=0 1 i −inτ
αe
cos nσ,
nn 1 a −inτ
αe
sin nσ.
nn (2.3.1.20)
(2.3.1.21) The mode expansions for the NS sector fermions look as follows
i
ψ− = 1
√
2 i
ψ+ = 1
√
2 a
ψ− = 1
√
2 − bi e−irσ ,
r (2.3.1.22) r ∈Z+ 1
2
+ bi e−irσ ,
r (2.3.1.23) r ∈Z+ 1
2
− ba e−irσ ,
r (2.3.1.24) r ∈Z+ 1
2 1
a
ψ+ = − √
2 + bae−rσ .
r (2.3.1.25) 1
r ∈Z+ 2 For the R sector fermions, one obtains
i
ψ− =
i
ψ+ =
a
ψ− = − di e−inσ ,
n
n∈Z
n∈Z 33 + di e−inσ ,
n (2.3.1.27) − da e−inσ ,
n n∈Z a
ψ+ = − (2.3.1.26) n∈Z (2.3.1.28)
+ da e−inσ .
n (2.3.1.29) If we had chosen the open string half as long as the closed one we would not need this replacement. 2. Open strings 66 The next step is to eliminate two directions by performing the light cone gauge. We
take this to be the timelike (Neumann) direction and a spacelike Neumann direction,
which we choose to be x1 .34 For the open string we have only an NS sector and
an R sector. Since the right movers are not independent from the left movers, the
right and left moving sectors do not decouple anymore. The constraints that the
expressions (2.1.1.24) – (2.1.1.27) vanish are not all independent. The zero mode part
of vanishing energy momentum tensor again yields the mass shell condition (since the
mode expansions diﬀer by factors of two, there is a diﬀerence of a factor of four as
compared to the closed string (2.1.2.42)),
M 2 = 2 (N − a) , (2.3.1.30) where a is the normal ordering constant and the number operator N is deﬁned in
(2.1.2.43) for the NS sector and in (2.1.2.61) for the R sector. In the NS sector the
GSO projection operator is as in (2.1.2.50) with the second factor removed. The lowest
GSO invariant states in the NS sector are
bi 1 k
−
2 , ba 1 k ,
− (2.3.1.31) 2 where we have indicated again the momentum eigenvalue of the vacuum by k. The
ﬁrst set of these states transforms in the vector representation of SO(p − 1) – the little
group of the unbroken Lorentz group. Hence, this state should be massless leading to
the consistency condition35
1
aN S = .
2 (2.3.1.32) Like in the closed string, this translates into a condition on the number of targetspace
dimensions
d = 10. (2.3.1.33) The ﬁrst states in (2.3.1.31) (with label i) form a U (1) gauge ﬁeld. The states with
label a are scalars transforming in the adjoint of U (1) (here, this appears just as a
pompous way of saying that they are neutral under U (1), however, later we will discuss
34 We do not consider an open string ending on a D0 brane, here. As in the case of the closed
string, it is useful to take Lorentz invariance as a guiding principle for a consistent quantization. For
a D0 brane, the Lorentz group is broken down to time reparameterizations which is too small for our
purposes. Later, we will see that we can obtain the D0 brane by Tdualizing a higher dimensional
Dbrane.
35
In principle, we could combine the ﬁrst states in (2.3.1.31) with one of the second states in order
to form a massive vector representation as long as p < d − 1. However, later we will see that the case
p = d − 1 is related by Tduality to the other cases. With this additional ingredient it follows that
the states in (2.3.1.31) must be massless. 2. Open strings 67
Spin/ PHOTON 1 0 1 M 2α Figure 2.6: NS mass spectrum of the open superstring
nonabelian gauge groups where those ﬁelds are adjoints rather than singlets). Since
the center of mass position of the open string is conﬁned to be within the world volume
of the Dbrane, all the open string states correspond to targetspace particles which
are conﬁned to live on the Dbrane. The NS massspectrum is depicted in ﬁgure 2.6.
The construction of the R sector vacuum state goes along the same lines as in
the closed string. The ten dimensional Majorana spinor decomposes into a couple of
spinors with respect to the unbroken Lorentz group SO(p, 1). We impose the GSO
projection by multiplying the states with one of the projection operators deﬁned in
(2.1.2.59). The sign is a matter of convention. The Rvacuum is massless on its own.
It leads to target space spinors providing all fermionic degrees of freedom which are
needed to obtain the maximal rigid supersymmetry in p + 1 dimensions.36
In the following sections we will investigate systems with more than one Dbrane.
This will lead to nonabelian ﬁeld theories on a stack of Dbranes. But before doing
so, we will brieﬂy discuss the possible Dbrane setups which are in agreement with
supersymmetry.
2.3.1.3 Number of ND directions and GSO projection At ﬁrst, consider the case that we have an open string with an odd number of ND
directions. Thus, we will have an odd number of directions where the worldsheet
36 The maximally possible amount of supersymmetry diﬀers for rigid and local supersymmetry. In
rigid supersymmetry, the highest occurring spin should not exceed one, whereas in locally supersymmetric ﬁeld theories, spin two ﬁelds (the gravitons) are allowed. In 3 + 1 dimensions, the maximal
supersymmetry is N = 4 (N = 8) for rigid (local) supersymmetry. From this one can deduce the
maximally allowed amount of supersymmetry in higher dimensions by viewing the 3 + 1 dimensional
theory as a toroidally compactiﬁed higher dimensional theory. 2. Open strings 68 fermions have zeromodes. For example in the R sector, the zeromodes form a Cliﬀord
algebra in p + 1 = odd dimensions. The representation of this algebra by the R ground
state will be irreducible (there is no notion of chirality in odd dimensions). Therefore,
we cannot perform the GSO projection on those states. The theory will not lead to
targetspace supersymmetry.
Let us now discuss the case of an even number of ND directions, taken to be 8 − 2n.
Then the GSO projection operator on the R sector ground state will be of the form
PGSO = 1 ± 2n d2 . . . d2n+1 .
0
0 (2.3.1.34) Using some algebra this can be written as
PGSO = 1 ± eiπ(J23 +...+J2n,2n+1 ) , (2.3.1.35) where the Jkl are the generators of rotations in the kl plane
Jkl = − ikl
d ,d .
200 (2.3.1.36) The eigenvalue of the ND Ramond groundstate under a 180◦ rotation in one plane is
±i. Thus, the eigenvalues of the R groundstate R under PGSO will be
PGSO R = (1 ± in ) R . (2.3.1.37) From this we deduce that the GSO projection is possible only if the number of ND
directions is an integer multiple of four. This means for example that, if a lower
dimensional Dbrane lives inside the worldvolume of a higher dimensional Dbrane,
the higher dimensional Dbrane has to extend in four or eight more directions. We
could have deduced this result faster by noting that (2.3.1.34) deﬁnes a projection
operator only if n is a multiple of four since otherwise the second term in (2.3.1.34)
squares to −1.
2.3.1.4 Multiple parallel Dbranes – Chan Paton factors In this section we will discuss sets of parallel Dpbranes.37 First, let us have a look
at two parallel Dpbranes which are separated by a vector δca in the transverse space.
(Later we will see that the distance between parallel Dbranes is a modulus, i.e. any
value is consistent.) From strings ending with both ends on the same Dbrane we
obtain the same spectrum as discussed in the section 2.3.1.2. In particular, we obtain
a U (1) × U (1) gauge symmetry where the corresponding gauge ﬁelds live on the ﬁrst
brane for the ﬁrst U (1) and on the second brane for the second U (1) factor.
37 Recall that the worldvolume of a Dpbrane has p space like and one time like dimension. 2. Open strings 69 In addition, we have strings stretching between the two branes. There are two
such strings with opposite orientations. As compared to section 2.3.1.2, only the
mode expansion for the bosons in Dirichlet directions is modiﬁed. The string starting
on the brane at ca and ending on the brane at ca + δca has the mode expansion
X a = ca + δca
σ−
π n=0 1 a −inτ
αe
sin nσ.
nn (2.3.1.38) The string with the opposite orientation is obtained by replacing σ → π − σ . We
rewrite the term with δca in a suggestive way
δca σ = 1a+
δc σ − σ −
2 (2.3.1.39) and compare with the expressions (2.1.5.9) and (2.1.5.10). The ﬁnite distance between
the Dbranes enters the mode expansion in a very similar way as the winding number
in the toroidally compactiﬁed closed string does. This is also intuitively expected – the
winding closed string is stretched around a compact dimension. As a ﬁnite winding
number also the ﬁnite distance contributes to the mass of the stretched string state,
it results in a shift of
δM 2 = (δca)2
.
π2 (2.3.1.40) The strings stretching between the branes transform under U (1) × U (1) with the
charges (1, −1) and (−1, 1) depending on the orientation. We will see below that
these charge assignments are necessary for consistency. Pictorially, they are obtained
by the rule that a string starting at a brane has charge +1 with respect to the U (1)
living on that brane whereas it has charge −1 if it ends on the brane. (The photon
which starts and ends on the same brane has net charge zero.) The U (1) × U (1) can
be also rearranged into a diagonal and a second U (1) such that all states are neutral
under the diagonal U (1).
The lightest GSOeven states in the NS sector of the stretched string are
i
ψ− 1 12
2 a
ψ− 1
2 12 i
, ψ− 1 21 , (2.3.1.41) 2 , a
ψ− 1
2 21 (2.3.1.42) where 12 and 21 denote the NS vacua for strings stretched between the two Dbranes and we have dropped the zero mode momentum eigenvalues k in the notation.
In this sector, the lightest states form a vector and d − (p +2) scalars. (Note that, in the
light cone gauge, we have to combine one of the transverse excitations (2.3.1.42) with
the longitudinals (2.3.1.41) in order to obtain a massive vector.) The R sector states
provide the fermions needed to ﬁll up supermultiplets. (The amount of supersymmetry
is the same as in the single brane case.) 2. Dbrane interactions 70 Now, take the interbrane distance to zero. We obtain two massless vectors and
2 (d − p − 1) massless scalars. Together with the massless ﬁelds coming from strings ending on identical branes, the vectors combine into a U (2) gauge ﬁeld, and the scalars
combine into d − p − 1 scalars transforming in the adjoint of U (2). (One can split
U (2) into a diagonal U (1) times an SU (2). All ﬁelds are SU (2) adjoints and neutral
under U (1).) Moving the Dbranes apart from each other can be viewed as a Higgs
mechanism from the target space perspective. The amount of supersymmetry leads
to ﬂat directions for the scalars in the adjoint of U (2). This means that a scalar can
have some non zero vev breaking U (2) to U (1) × U (1). With the given amount of supersymmetry, all massless ﬁelds transform in the same representation of the gauge
group as the vector bosons (viz. in the adjoint). Therefore, the Higgs mechanism can
only work for nonabelian gauge groups. Our charge assignments of the open strings
stretched between two diﬀerent Dbranes thus lead to a consistent picture.
An economic way of studying systems with N parallel Dbranes is to replace all
the diﬀerent sectors corresponding to the possibilities of strings stretching among the
N Dbranes by one matrix valued state
· → ·, ij λji (2.3.1.43) where λ is an N × N matrix. The component λji corresponds to a string stretching between the ith and the j th brane. The matrix λ is called ChanPaton factor. Consider
again the case where all the N Dbranes are separated in the transverse space. For
the lightest NS sector states the diagonal elements λii are N U (1) gauge ﬁelds and
d − p − 2 scalars. They are neutral under U (1)N , i.e.
λii = λ† .
ii (2.3.1.44) The oﬀdiagonal elements correspond to massive vectors and scalars. Open string
sectors with opposite orientation have opposite charges under U (1)N , i.e.
λij = λ† .
ji (2.3.1.45) The ChanPaton factor is a unitary N × N matrix. Maximal gauge symmetry is ob tained when all N Dbranes sit at the same point in the transverse space. The diagonal
and oﬀdiagonal elements of λ combine and give rise to a U (N ) vector multiplet. 2.3.2 Dbrane interactions Already at an intuitive level, one can deduce that Dbranes interact. This comes about
as follows. The two ends of an open string ending on the same Dbrane can join to
form a closed string. The closed string is no longer bound to live on the Dbrane, 2. Dbrane interactions 71 τ
σ Di Dj Figure 2.7: Dbrane Di and Dbrane Dj talking to each other by exchanging a closed
string.
it can escape into the bulk of the target space. In particular, it may reach another
Dbrane by which it is absorbed. The absorption process is inverse to the emission
process. The closed string hits the Dbrane where it can split into an open string
which is constrained to live on the second Dbrane. Dbranes talk to each other by
exchanging closed strings. In ﬁgure 2.7 we have drawn such a process. In order to
make contact to conventions of the standard reviews on Dbrane physics we take the
closed string twice as long as the open string (see also the footnote 33). This implies
±
±
that in the closed string mode expansions we replace e−2inσ → e−inσ .
We will compute the process depicted in ﬁgure 2.7 in Euclidean worldsheet signature. The range of the worldsheet coordinates is
0 ≤ σ < 2π , 0 ≤ τ < 2πl. (2.3.2.1) The Euclidean worldtime τ is taken to be compactiﬁed on a a circle of radius 2l. The
worldtime taken by a string to get from one brane to the other one is 2πl – this process
can be periodically continued such that one period lasts 4πl. (The factor of 2π is a
matter of convention. It is introduced because compact directions are usually speciﬁed
by the radius of the compactiﬁcation circle rather than its circumference.) Note also,
that l has nothing to do with the distance of the Dbranes. The distance in target
space will appear later and will be denoted by y .
Since we have deﬁned the Dbranes in terms of open strings it will be useful to
compute also the Dbrane interactions in terms of open strings. To this end, we
perform a so called worldsheet duality transformation, i.e. we interchange σ with τ .
The resulting picture is an open string oneloop vacuum amplitude. It is described by
the annulus diagram drawn in ﬁgure 2.8. 2. Dbrane interactions 72 τ σ σ Dj Di Figure 2.8: Dbrane Di and Dbrane Dj talking to each other by a pair of virtual open
strings stretching between them.
The parameter ranges for the open string are
0 ≤ τ < 2π , 0 ≤ σ < 2πl. (2.3.2.2) The periodicity of closed string worldsheet fermions is related to the behavior of
open strings under shifts in τ by 2π . The diagram 2.8 corresponds to a vacuum amplitude and thus to a trace in the open string sector. This trace is actually a supertrace
with respect to worldsheet (and target space) supersymmetry. The additional sign in
the trace over worldsheet fermions is imposed by specifying the boundary condition
µ
µ
ψ± (τ + 2π, σ ) = (−)F ψ± (τ, σ ) . (2.3.2.3) Thus, a (−)F insertion (canceling the (−)F in (2.3.2.3)) corresponds to closed string RR sector exchange whereas no (−)F insertion yields the closed string NSNS sector
exchange. From the open string perspective there is no exchange of NSR or RNS
sector closed strings between the Dbranes. In the picture 2.7, the Dbrane appears as a boundary state of the closed string.38 This boundary state is a superposition
of an NSNS sector state and an RR sector state. There are no NSR or RNS sector
contributions. This can be explained by the fact that the Dbrane is a target space
boson. It is speciﬁed by the target space vector ca in (2.3.1.5) and hence transforms
as a vector and not as a spinor under rotations in the space transverse to the brane.
We will not present the details of the boundary state formalism here, and recommend
38 It will turn out that the closed strings which are exchanged in ﬁgure 2.7 are type II superstrings. 2. Dbrane interactions 73 the review [195] instead. To be slightly more speciﬁc let us just present the deﬁning
equation for a boundary state in closed string theory (as usual we label the Neumann
directions by i = 0, . . . , p and the Dirichlet directions by a = p + 1, . . . , 9)
∂τ X i Dbrane = (X a − ca) Dbrane = 0. (2.3.2.4) This relates the right moving and left moving bosonic excitations the boundary state
can carry. Applying the worldsheet supersymmetry transformations (2.1.1.22) and
(2.1.1.23) on (2.3.2.4) and requiring that there is a combination of the two supersymmetries which annihilates the boundary state tells us that the boundary state should
have the same number of right moving and left moving fermionic excitations. Also,
when + is taken to be (anti)periodic then − should have the same periodicity. The
boundary state cannot be excited by NS fermions in, say, the right moving sector and
R fermions in the left moving sector. It has only an NSNS and an RR sector.
Instead of the nonstandard range for the open string worldsheet coordinates we
would like to have the standard range
0 ≤ τ < 2πt , 0 ≤ σ < π.
In order to achieve this we redeﬁne τ → τ t and σ → (2.3.2.5)
σ
2l . Under this redeﬁnition, the Hamiltonian (which is obtained by integrating the τ τ component of the energy
momentum tensor over σ ) transforms according to
H → 2lt2H. (2.3.2.6) Further, we want the time evolution operator when going once around the annulus
to transform as (2πt should be identiﬁed with the worldsheet time it takes the open
string to travel around the annulus once)
e−2πH → e−2πtH . (2.3.2.7) 1
tl = .
2 (2.3.2.8) This yields the relation The annulus vacuum amplitude in ﬁgure 2.8 yields the vacuum energy of an open string
starting on the Dbrane Di and ending on the Dbrane Dj . This can be expressed as
1
1
dH −
1
=
− log det H = − tr log H = lim tr
2
2
2 →0
d
∞
1
d
dt −2πtH
1
+ log 2π − Γ (1) .
tr lim
te
→0 d
2t
2
2
0 (2.3.2.9) 2. Dbrane interactions 74 At a formal level this expression is correct. However, the next step is to take the
limit of → 0 before performing the integral over t. This would be allowed only if the integral were converging. This is not the case in most of the applications (for
example, the integral diverges if H is just a number and no trace is taken). But the
error done is some unknown additive constant contribution which is not of interest for us39. Together with this unknown constant we also drop the
obtain for the amplitude in ﬁgure 2.8 (reinstalling α )
∞
0 1
2 (log 2π − Γ (1)) and dt
Str e−2πα tH
2t (2.3.2.10) (here we have replaced the trace by a supertrace. It refers to target space supersymmetry, i.e. the trace receives an additional minus sign for target space spinors. The
integral over t is usually regulated by a UV cutoﬀ near t = 0.) The expression (2.3.2.10)
has also an intuitive interpretation. The supertrace describes a process where a pair of
open strings is created from the vacuum, then propagates for a time 2πt and annihilates. This corresponds to the diagram drawn in ﬁgure 2.8. Further, we integrate over
dt
all possible moduli t of the annulus with the measure 2t . The Hamiltonian is p2 + M 2
which can be expressed by use of (2.3.1.30) and (2.3.1.40) as follows
H = p2 + y2
+ 2 (N − a) ,
π2 (2.3.2.11) where y is the distance between the two Dbranes, and a is the normal ordering
1
1
constant ( 24 per bosonic direction, 48 per fermionic direction in the NS sector, and
1
− 24 per fermionic direction in the R sector). Recalling that in this expression we 1
have set α = 2 gives (just multiply with appropriate powers of 2α to get the mass
dimension right) α H = α p2 + y2
+ (N − a) .
4π 2α (2.3.2.12) It is useful to split (2.3.2.10) into several contributions
dt
Str e−2πα tH =
2t
dt
tr ZERO e−2πtα H0 trBOSONS e−2πtHB
2t MODES
trGSO
NS
FERMIONS e−2πtHN S − trGSOR e−2πtHR . (2.3.2.13) FERMIONS We have split the Hamiltonian into
α H = α H0 + HB + HN S/R,
39 (2.3.2.14) In our case the trace is actually a supertrace which vanishes when taken over a constant. However,
since the corresponding series does not converge absolutely the result depends on the ordering in which
we take the trace. 2. Dbrane interactions 75 with
H0 = p2 + y2
,
4α 2 π 2
∞ 8 HB =
i=1
8 HN S =
i=1
8 HR =
i=1 n=1 ∞ r= 1
2
∞ n=1 (2.3.2.15) αi n αi −
−
n 1
24 rbi r bi −
−r 1
,
48 ndi n di +
−n , (2.3.2.16) 1
24 (2.3.2.17) . (2.3.2.18) An additional minus sign in the R sector contribution is due to the fact that we take
the supertrace with respect to space time supersymmetry (Rsector states are space
time fermions). The superscript GSO indicates that the trace is taken over GSO even
states. We will clarify this point later.
The trace over the zero modes is
tr ZERO
MODES = 2Vp+1 dp+1 k −2πtα k2 − ty2
2
2πα = 2Vp+1 8π α t
e
(2π )p+1 − p+1
2 ty 2 e− 2πα , (2.3.2.19) where the factor of two counts the possible orientations of the open string traveling
through the annulus. The factor Vp+1 denotes formally the worldvolume of the parallel
Dp branes. It arises due to the normalization of states with continuous momentum
( pp = δ (p+1) (0) = Vp+1 / (2π )p+1 ). To express oscillator traces, it is useful to deﬁne
the following set of functions40
1 f1 (q ) = q 12
1 f3 (q ) = q − 24 ∞ n=1
∞ 1 − q 2n 1 + q 2n−1 1 , f2 (q ) = q 12 √
2
1 , f4 (q ) = q − 24 n=1 ∞ 1 + q 2n , n=1
∞
n=1 1 − q 2n−1 . (2.3.2.20) These functions satisfy the identity
8
8
8
f3 (q ) = f2 (q ) + f4 (q ) . (2.3.2.21) In order to translate the open string calculation back to the closed string process
(ﬁgure 2.7) we will make use of the modular transformation properties,
π f1 e− s
40 = √ π sf1 e−πs , f3 e− s π = f3 e−πs , f2 e− s = f4 e−πs . (2.3.2.22) These are related to the Jacobi theta functions, which are also often used in the literature. 2. Dbrane interactions 76 Next, we are going to compute the trace over the worldsheet bosons. The sum over the
coordinate label i in (2.3.2.16) can be written in front of the exponential as a product.
Nothing depends explicitly on the direction i, therefore this gives a power of eight to
the result for a single bosonic direction. The second sum can be decomposed into a
level part and an occupation number part, giving the result
trBOSONS e −2πtHB = e ∞ πt
12 ∞ 8
−2πtlk e . (2.3.2.23) l=1 k =0 Here, l denotes the level of a creator α−l acting on the ground state and k is the
occupation number (the number of times this creation operator acts). The sum over
k is just a geometric series, and we obtain the result
trBOSONS e−2πtHB = 1
.
8
f1 (e−πt ) (2.3.2.24) The calculation of the traces over the fermionic sectors is similar. Let us just point
out the diﬀerences. First of all, we have to take the trace only over GSO even states.
This is done by inserting the GSO projection operator into the trace and summing
over all states
1
1
trGSO (· · · ) = tr (· · · ) + tr (−)F · · · .
2
2 (2.3.2.25) The second diﬀerence – as compared to the bosonic calculation – is that for worldsheet fermions the occupation number can be only zero or one (since the creators
anticommute). The NS trace without the (−)F insertion comes out to be
18
1
trNS e−2πtHN S = f3 e−πt .
2
2 (2.3.2.26) Since the NS vacuum is GSO odd we assign an additional minus to states with even
and zero occupation if (−)F is inserted into the trace,
1
18
trNS (−)F e−2πtHN S = − f4 e−πt .
2
2 (2.3.2.27) For the Rsector trace without the (−)F insertion one obtains
1
18
− trR e−2πtHR = − f2 e−πt ,
2
2 (2.3.2.28) where the 16fold degeneracy of the R vacuum has been taken into account by the
√
factor of 2 in the deﬁnition of f2 (2.3.2.20). The R sector trace with a (−)F insertion
vanishes identically. Half of the R sector groundstates have eigenvalue +1 whereas the
other half has eigenvalue −1. Adding up all the results and using the identity (2.3.2.21) we ﬁnd that the net result for the annulus amplitude vanishes. This implies also that 2. Dbrane interactions Sector 77 l→∞ dl× RR 1
 2 4α π 2 NSNS 1
2 4α π 2 − p+1
2 − p+1
2 Vp+1 l Vp+1 l p−9
2 y2 e− 4πα l
y2 p−9
2 e− 4πα l Table 2.7: Contributions of massless RR and NSNS sector closed strings to ﬁgure 2.7.
the closed string diagram 2.7 vanishes, and a hasty interpretation of this would lead
to the conclusion that Dbranes do not interact (at least not via the exchange of
closed strings). However, as we will argue now, this is not the case. The situation
rather is that repulsive and attractive interactions average up to zero. In order to
see this41, let us translate the annulus result back to the tree channel. Further, we
would like to ﬁlter out the contributions of massless closed string excitations. To this
end, we replace t in terms of l using (2.3.2.8), and afterwards apply (2.3.2.22). The
contribution of massless closed string excitations is obtained by focusing on the leading
contribution in the l → ∞ limit. (Massless interactions have inﬁnite range whereas
the interactions carried by massive bosons have ﬁnite range.) We collect the result
of this straightforward calculation in table 2.7. We separate closed RR contributions
from NSNS sector contributions. The former ones correspond to the (−)F insertion
and the latter to the 1 insertion in the annulus amplitude.
From table 2.7 we deduce that interactions carried by closed strings in the RR
sector cancel interactions mediated by closed strings in the NSNS sector. One can
take the diagram 2.7 to the ﬁeld theory limit. In that limit the ‘hose’ connecting
the two Dbranes becomes particle propagators (lines). In the NSNS sector we ﬁnd
propagators for the metric (ﬂuctuations), the dilaton and the antisymmetric tensor
Bµν . The Dbranes appear as source terms for those ﬁelds. The NSNS contribution to
diagram 2.7 tells us the strength of this coupling. In particular, it yields the strength
of the gravitational coupling which is given by the tension Tp. A detailed analysis of
the eﬀective ﬁeld theory and comparison with table 2.7 leads to the result [371]
2
Tp =
41 π
4π 2α
κ2 3−p −2Φ0 e , (2.3.2.29) Already in the annulus computation, there are signs for such a cancellation. For the result, the
minus sign in front of the R sector contribution was essential. Since R sector states are target space
fermions, this indicates that the result is due to target space supersymmetry. 2. Dbrane actions 78 where κ is the gravitational coupling in the eﬀective theory (see section 2.1.4) and
Φ0 denotes the constant vev of the dilaton. Even though we did not derive this
explicitly here, let us make a few comments to motivate the expression qualitatively.
In a ﬁeld theory calculation the propagator of an NSNS ﬁeld is accompanied by a
power of κ2e2Φ0 . With Tp deﬁned as in (2.3.2.29) the κ and Φ0 dependence drop
out. Since κ2 ∼ (α )4 , the mass dimension of Tp is correct. (The α dependence
in the string calculation yields agreement with (2.3.2.29) after substituting for the
integration variable l such that the α dependence in the exponent vanishes.) Further,
the exchange of massless particles should lead to a Coulomb interaction in ﬁeld theory.
For the interaction between the two Dbranes this means that the potential should be
given by the distance to the power of two minus the number of transverse dimensions
V ∼ y p−7 . (2.3.2.30) In the string result (table 2.7), we can extract the y dependence after rescaling the
integration parameter l such that the y dependence in the exponent disappears. The
result agrees with (2.3.2.30). In order to ﬁx the numerical coeﬃcient, one needs to do
a more detailed analysis of the ﬁeld theory calculation. More details on this can be
found in Polchinski’s book[371].
The second line in table 2.7 tells us that and how string RR ﬁelds couple to the
brane. We ﬁnd the same Coulomb law as for the gravitational interaction. The RR
ﬁeld should be a p + 1 form. The p = even branes interact via closed type IIA strings
and the p = odd branes via closed type IIB strings. The value of the RR contribution
is exactly minus the value of the NSNS contribution. This provides us with the RR
charge of the Dbrane42
2
µ2 = 2κ2Tp e2Φ0 ,
p (2.3.2.31) where we have taken into account the dilaton dependent RR ﬁeld redeﬁnition performed in section 2.1.4. The signs are undetermined at this level. 2.3.3 Dbrane actions In the following we will specify the actions for the ﬁeld theory on the Dbrane. We
will also argue that the Dbrane interaction with the bulk ﬁeld is obtained by adding
the action for ﬁelds living on the Dbrane (the Dbrane action) to the eﬀective type
II action of section 2.1.4. The previous calculation ﬁxes the coeﬃcient in front of the
Dbrane action.
42 The factor of 2κ2 has been introduced in order to match the charge deﬁnition to be used later in
equ. (3.3.0.12). 2. Dbrane actions
2.3.3.1 79 Open strings in nontrivial backgrounds In this section we will modify the calculation presented in section 2.1.3 such that
it accommodates open string excitations. We perform a Wick rotation such that the
worldsheet is of Euclidean signature. Further, we map the parameter space of the open
string worldsheet from a strip to the upper half plane via the conformal transformation
z = eτ +iσ = z 1 + iz 2 . (2.3.3.1) The discussion is performed for bosonic strings but later we will also give the result for
superstrings. Since the open string couples naturally to closed strings (via joining its
ends) we also switch on nontrivial closed string modes. These are the metric Gµν and
the antisymmetric tensor Bµν . We will take those background ﬁelds to be constant.
For the open string we take Neumann boundary conditions in all directions at ﬁrst.
(For the superstring this is not consistent with RR conservation. At the moment we
will ignore this problem and return to it later.) Later, we will discuss that Tduality
maps Neumann to Dirichlet boundary conditions. Hence, the restriction to Neumann
boundary conditions will not result in a loss of generality. At ﬁrst, we consider a single
brane setup. The massless open string excitation is now a U (1) gauge ﬁeld Aµ . We
restrict ourself to the special case that the U (1) ﬁeld strength Fµν is slowly varying, i.e.
we neglect contributions containing second or higher derivatives of Fµν . Under these
conditions we will be able to perform the calculation to all orders in α (in diﬀerence
to section 2.1.3). The nonlinear sigma model reads
S= 1
2πα d2 z +i
z 2 =0 1
∂α Xµ ∂ α X µ + i
2 αβ Bµν ∂α X µ ∂β X ν dz 1Aµ ∂1X µ . (2.3.3.2) Here, A has been rescaled such that α appears as an overall factor in front of the action.
The target space indices µ,ν are raised and lowered with the constant background
metric Gµν . The worldsheet metric is taken to be the identity in the z α coordinates
(2.3.3.1).43 Using Stoke’s theorem the term with the constant B ﬁeld can be rewritten
as a surface integral
d2z αβ Bµν ∂α X µ ∂β X ν = 2 z 2 =0 dz 1 Bµν X µ ∂1 X . (2.3.3.3) The term with the Bﬁeld can be absorbed into a redeﬁnition of the gauge ﬁeld Aµ
and we can put it to zero without loss of generality. (It can be recovered by replacing
F → F − 2B .)
43 In principle this choice introduces gauge ﬁxing ghosts as stated in section 2.1.3. Since their eﬀect
is not altered by the presence of the boundary we will not discuss the ghosts here. We should, however,
mention that there are technical subtleties when taking into account the dilaton in worldsheets with
boundaries, see e.g. [49]. (Recall that the ghosts contribute to the dilaton beta function.) 2. Dbrane actions 80 In order to proceed we specify a classical conﬁguration around which we are going
¯
to expand. We denote this again by X µ . For freely varying ends the equation of
motion and boundary conditions read
¯
∂z ∂z X µ = 0,
¯
¯
¯
∂2 X µ + iFν µ ∂1 X ν z 2 =0 (2.3.3.4) = 0. (2.3.3.5) The presence of the U (1) gauge ﬁeld Aµ results in inhomogeneous Neumann boundary
conditions. Since we have restricted ourselves to the case where the target space
metric is constant, the background ﬁeld expansion simpliﬁes in comparison to the
computation of section 2.1.3. The ﬁelds can simply be Taylor expanded. Neglecting
second and higher derivatives of Fµν , the background ﬁeld expansion terminates at
the third order in the ﬂuctuations, + i
2πα dz 1
z 2 =0 1
2πα 1
d2z ∂ α ξµ ∂α ξ µ
2
1
1
1
¯
Fµν ξ ν ∂1 ξ µ + ∂ν Fµλ ξ ν ξ λ ∂1 X µ + ∂ν Fµλ ξ ν ξ λ ∂1 ξ µ . (2.3.3.6)
2
2
3 ¯
¯
S X +ξ = S X + Since we have chosen the worldsheet metric to be the identity (and the geodesic curvature of the boundary to vanish) a suitable technique to integrate out the ﬂuctuations
ξ µ is given by a Feynman diagrammatic approach. This means that we split the action
into a free and an interacting piece. The free piece determines the propagator whereas
the interacting one leads to vertices. As the free part of the action we take
Sf ree = 1
4πα d2 z∂ αξµ ∂α ξ µ + i
4πα z 2 =0 dz 1Fµν ξ ν ∂1 ξ µ . (2.3.3.7) Hence, the interacting part is given by the rest
Sint = i
2πα dz 1
z 2 =0 1
1
¯
∂ν Fµλ ξ ν ξ λ ∂1 X µ + ∂ν Fµλ ξ ν ξ λ ∂1 ξ µ .
2
3 (2.3.3.8) In order to compute the propagator we have to invert the two dimensional Laplacian
and to satisfy the (inhomogeneous Neumann) boundary conditions arising from the
variation of Sf ree with respect to ξ µ (with free varying ends of the ξ µ ), ∂2 ∆µν 1
2∆µν z , z
2πα
z , z + iFµ λ ∂1 ∆λν z , z z 2 =0 = −δ z − z Gµν ,
= 0. (2.3.3.9)
(2.3.3.10) This boundary value problem can be solved (for example by borrowing the method of
mirror charges from electro statics) with the result
∆µν z , z = −α Gµν log z − z 
ˆ
+ G−1
z − z 
¯ µν ¯
log z − z 2 + θµν log z −z
¯
.(2.3.3.11)
z −z
¯ 2. Dbrane actions 81 µ ν z z ν ∆µν (z, z ) λ
1
¯µ
2 ∂ν Fµλ ∂1 X µ ν λ
1
3 ∂ν Fµλ · (∂1 ·)· Figure 2.9: Feynman rules: The dotted line denotes the worldsheet boundary. The
slash on the leg means that a derivative acts on the corresponding leg.
Here, we have introduced the following matrices (an index S (A) stands for (anti)symmetrization and Gµν are the components of the inverted target space metric G−1 ,
as usual)
ˆ
G−1
ˆ
G µν µν θµν = 1
G+F µν =
S = Gµν − F G−1 F
= 1
G+F µν =
A 1
1
G
G+F G−F
µν µν , , (2.3.3.12)
(2.3.3.13) 1
1
F
G+F G−F µν . (2.3.3.14) The interaction piece Sint gives rise to two vertices —one with two and one with
three legs— located at the boundary. The Feynman rules are summarized in ﬁgure
2.9.
The propagator ∆µν (z, z ) becomes logarithmically divergent if the arguments coincide. Therefore, we replace the logarithm of zero by its dimensionally regularized
version
log z − z z =z =− d2k k(z−z
e
2k2 ) z =z = − lim µ
→0 d 2− k
,
2 (k2 + m2 )
(2.3.3.15) where the momentum integral extends over a two dimensional plane. We introduced a
mass scale µ which is needed in order to keep the mass dimension ﬁxed while changing
the momentum space dimension. In the last step we have introduced also an infrared 2. Dbrane actions 82 cutoﬀ m2 .44 With our regularization prescription we obtain
log z − z z =z 1 2−
= π2
2 µ
m Γ 2 , (2.3.3.16) which has a simple pole as goes to zero.
The bare background ﬁeld (coupling) Aµ is inﬁnite. By adding counterterms to
the action the bare ﬁeld can be expressed in terms of a renormalized ﬁeld which is
ﬁnite as goes to zero. The only counterterm arises from the diagram in ﬁgure 2.10.
The action is written in terms of renormalized ﬁelds by adding
δS = − i
2πα z 2 =0 1
¯
dz 1 ∂ν Fµλ ∂1 X µ ∆νλ z 1 , z 1
2 z →z . (2.3.3.17) (and replacing the bare gauge ﬁeld by the renormalized one. Hoping that the renormalization program is suﬃciently familiar we do not introduce sub or superscripts
indicating the diﬀerence between bare and renormalized couplings). The betafunction
d
of Aµ is obtained by applying µ dµ on the renormalized couplings and using the fact
that the bare couplings are independent of the cutoﬀ. This leads to (now, Aµ denotes
the renormalized coupling)
A
βρ = µ d
ˆ
Aρ = ∂ν Fρλ G−1
dµ λν . (2.3.3.18) In this case the vanishing of the beta function ensures conformal invariance. (We do
not encounter the subtleties which we met in section 2.1.3. Partially, this is the case
because we have written the action always in a manifestly gauge invariant form, i.e.
in terms of the gauge ﬁeld strength. By performing partial integrations diﬀerently we
could have carried out the calculation in a slightly more complicated way, with the
same result.) The equation of motion for the gauge ﬁeld is
A
βµ = 0. This equation of motion can be lifted to the DiracBornInfeld action
√
3−p
π
4π 2α 2
dp+1x e−Φ det (G + F ),
S=
κ (2.3.3.19) (2.3.3.20) where p + 1 is the number of Neumann directions (i.e. for our discussion p + 1 = 10(26)
for the super (bosonic) string45). The factor in front of the integral in (2.3.3.20) has
not been ﬁxed by our current discussion. We will explain how to ﬁx it below. The
same applies to the dilaton dependence. (We discussed only the case of a constant
dilaton Φ.) Since we have rescaled Aµ by powers of α such that the α dependence 2. Dbrane actions 83 Figure 2.10: The logarithmically divergent Feynman diagram.
appears as an overall factor in (2.3.3.2), the α expansion of the action (2.3.3.20) is
not obvious. Performing rescalings such that Aµ has massdimension one (or zero)
shows that the α expansion is a power expansion in Fµν . Also, we note that the
propagator (2.3.3.17) contains higher orders in α . Alternatively, we could have chosen
a propagator satisfying homogenous Neumann boundary conditions at the price of
having an additional vertex operator. This additional vertex does not lead to a one
loop (leading order in α ) divergence since it is antisymmetric in its legs. The leading
order α equation is
∂µ F µν = 0. (2.3.3.21) Lifting this to an action would give (in the small α approximation)
S∼ √
dp+1xe−Φ −GF 2 , (2.3.3.22) where the Φ dependence has been taken such that the result coincides with the small
α expansion of (2.3.3.20). Expanding (2.3.3.20) in powers of F and keeping only terms
√
up to F 2 we ﬁnd in addition to (2.3.3.22) a contribution dp+1 xe−Φ −G. From a
ﬁeld theory perspective this is a tree level vacuum energy. So far, we did not properly
couple the open string excitations to gravity. We included the eﬀects of bulk ﬁelds
on the equations of motion for open string excitations, but we did not encounter a
back reaction, i.e. that the ﬁeld Aµ enters the equations of motion for the closed string
excitations. The reason is that the back reaction is an annulus eﬀect. We will not
present a detailed annulus calculation but sketch the result. (In principle we have
done the necessary computations in the previous section.) Since the beta functions
depend on local features (short distance behaviors) one would guess that for the beta
function it may not matter whether the worldsheet is an annulus or a disc. However,
the annulus may degenerate as depicted in ﬁgure 2.11.
44 A diﬀerent (maybe more elegant) way to deal with infrared divergences is discussed e.g. in [102],
see also the appendix of[228].
45
We view the result of our computation as a result for the bosonic modes of the superstring. Since
we did not specify the eﬀective action for the closed bosonic string the factor in front of (2.3.3.20) is
irrelevant for the bosonic string (see also the discussion of the Fischler Susskind mechanism below). 2. Dbrane actions 84 Figure 2.11: Fischler Susskind mechanism: The annulus degenerates into a punctured
disc as the inner circle shrinks to zero. This gives rise to a closed string counterterm
depending on the open string excitations.
This gives an additional short distance singularity. (The inner circle of the annulus
becomes short.) This singularity can be taken care of by adding counterterms to
the closed string action. The counterterms depend on open string modes. This in
turn leads to terms in the closed string beta functions which depend on the open
string modes. This process is known as the Fischler Susskind mechanism. The net
eﬀect is that we add the open string eﬀective action to the closed string eﬀective
action (2.3.3.20) and obtain the equations of motion by varying the sum. This is
the expected back reaction. (In particular the Einstein equation now contains the
energy momentum tensor of the open string modes.) After taking the back reaction
into account, the coeﬃcient in (2.3.3.20) does matter. In the previous section we
have computed the tension of the Dbrane (2.3.2.29). This ﬁxes the coeﬃcient and the
dilaton dependence46 as given in (2.3.3.20). According to our discussion in the previous
section, the presence of a Dbrane should also backreact on the RR background. We
could not see this in the present consideration since we did not take into account non
trivial RR backgrounds. (In fact, it is rather complicated to switch on nontrivial RR
backgrounds in the nonlinear sigma model.) We will come back to the discussion of
RR contributions to the open string eﬀective action below.
So far, we have studied the case of a single Dbrane. How is this discussion modiﬁed
in the presence of multiple Dbranes? We have focused on the case where we have only
Neumann boundary conditions. This means that multiple Dbranes must sit on top
of each other, simply because there is no space dimension left in which they could be
separated. The eﬀect of having more than a single brane is that the gauge ﬁeld Aµ is
a U (N ) gauge ﬁeld – it is a matrix. Calling the expression (2.3.3.2) an action does not
make much sense anymore since we would have a matrix valued action. Therefore, one
46 Note also that this dilaton dependence agrees with our general discussion in section 2.1.4. The
Euler number of the disc diﬀers by one from the Euler number of the sphere. 2. Dbrane actions 85 takes just the bulk part of the action (the ﬁrst line in (2.3.3.2)) and computes instead
of the partition function the Wilson loop along the string boundary[142],
W = tr P ei ∂M ˙
dtAµ X µ , (2.3.3.23) where we have denoted the boundary of the worldsheet by ∂M and chosen some t
to parameterize this curve. The letter P stands for the path ordered product. The
expectation value is computed with respect to the bulk action only. Now, it is problematic to get an expression containing all orders in α . The leading α contribution
to the beta function results in the YangMills equation
µF where µν = 0, (2.3.3.24) denotes a gauge covariant derivative. The eﬀective action in leading ap proximation can be obtained as follows. We expand (2.3.3.20) to ﬁrst order in F 2 .
We replace F 2 by trF 2 . In addition, we multiply the zeroth order term in F by the
number N of Dbranes (the tension is N times the tension of a single Dbrane). The
generalization of the DiracBornInfeld action (2.3.3.20) to nonabelian gauge ﬁelds is
a subject of ongoing research, see e.g. [449, 75].
2.3.3.2 Toroidal compactiﬁcation and Tduality for open strings In the previous section we have discussed the case of having Neumann boundary
conditions in all directions. This means that the Dbranes have been space ﬁlling
objects. In order to obtain results for Dbranes extending along less dimensions we
will discuss Tduality for open strings, now.
At ﬁrst, we focus on the case with trivial background ﬁelds. From section 2.1.5 we
recall that Tduality interchanges winding with momentum modes. For the open string
we have either winding or momentum modes in compact directions. A string with
DD boundary conditions along the compact dimension can have nontrivial winding
modes. Since the ends of the string are tied to the Dbrane it cannot unwrap. On
the other hand, the DD string does not have quantized Kaluza Klein momenta. The
Dbrane can absorb any momentum carried by the string in the compact direction.
For NN strings, opposite statements are true. If the string has Neumann boundary
conditions along the compact dimension, its ends can move freely in that direction –
it can continuously wrap and unwrap the compact dimension. On the other hand, the
string cannot transfer KaluzaKlein momentum to the Dbrane. The NN string has
nontrivial momentum modes. This consideration suggests that Tduality for open
strings interchanges Neumann with Dirichlet boundary conditions.
Let us substantiate these qualitative statements by studying the eﬀect of Tduality
on the mode expansions. For the Tduality transformation we use the “recipe” (2.1.5.31). 2. Dbrane actions 86 To be speciﬁc we choose the ninth direction to be compact, i.e.
x9 ≡ x9 + 2πR. (2.3.3.25) For the string with NN boundary conditions in the ninth dimension this implies that
the center of mass momentum is quantized
p9 = n
,
R (2.3.3.26) with n being an integer. There are no integer winding numbers in the case of NN
boundary conditions. We rewrite the mode expansion (2.3.1.20) in a suggestive way
9
9
X 9 = XR + X L ,
n− i
x9
9
+
σ+
XR =
2
2R
2
9
XL = n+ i
x9
+
σ+
2
2R
2 (2.3.3.27) n=0 n=0 1 9 −inσ−
αe
,
nn (2.3.3.28) 1 9 −inσ+
αe
.
nn (2.3.3.29) Applying the recipe (2.1.5.31), we obtain the mode expansion for the Tdual coordinate
X9 TDUALITY −→ n
˜
X9 = σ +
R n=0 1 9 −inτ
αe
sin nσ.
nn (2.3.3.30) This mode expansion is zero at σ = 0 and 2nR at σ = π , where (see (2.1.5.20))
R= 1
α
=.
2R
R (2.3.3.31) The interpretation is that the open string ends on a Dbrane located at x9 = 047.
The open string winds n times around a circle of radius R . It is rather obvious that
—starting from a DD string with mode expansion (2.3.3.30)— Tduality will take us
to an NN string with mode expansion (2.3.3.27), (the center of mass position depends
again on the way we distribute a constant between left and right movers). So, Tduality inverts the compactiﬁcation radius and interchanges Dirichlet with Neumann
boundary conditions. We leave it to the reader to verify that an investigation of the
worldsheet fermions and of ND directions is consistent with this picture.
In section 2.3.2 we have noticed that Dpbranes with p even (odd) interact via
the exchange of closed type IIA(B) strings. Our present observation that Tduality
along a compact direction interchanges Dirichlet with Neumann boundary conditions
implies that a Dpbrane with even p is mapped onto a Dqbrane with odd q, and vice
47 We could have obtained a diﬀerent position by distributing the center of mass position of the NN
string x9 asymmetrically among left and right movers. 2. Dbrane actions 87 versa (q = p ± 1). This goes along nicely with our earlier statement (section 2.1.5.4)
that Tduality along one circle interchanges type IIA with type IIB strings.
Finally, let us discuss Tduality for open strings in the presence of nontrivial
background ﬁelds. For the closed string we have done this in section 2.1.5.3. Because
the discussion of the closed string background ﬁelds is not aﬀected by the open string,
we will focus on the special case where only the open string gauge ﬁeld is nontrivial.
For simplicity we also restrict to one Dbrane only (for multiple Dbranes see e.g.
[143])48. Let us ﬁrst outline in words the procedure we are going to carry out. The
compactiﬁcation has to be done in a Killing direction. (Shifts along the compact
direction are isometries.) We will take this dimension to be the ninth. The next step
is to gauge this isometry and to undo the gauge by forcing the corresponding gauge
ﬁeld to be trivial. This will be done again by adding a Lagrange multiplier times the
ﬁeld strength of the isometry gauge ﬁeld. The Lagrange multiplier will become the
Tdual coordinate in the end. In particular the Lagrange multiplier lives on a circle
whose radius is inverse to the original compactiﬁcation radius. This is derived from
the requirement that the radius of the isometrygauge group (U (1)) agrees with the
compactiﬁcation radius. We will not discuss the technical details of this derivation
(they are presented for example in the appendix of [15]). Instead, we will focus on
a detailed discussion of the boundary conditions. The boundary condition of the
isometry gauge ﬁelds is constrained by the boundary condition of the open string. This
will be implemented by a second Lagrange multiplier which lives only at the boundary
of the worldsheet. After integrating out the isometry gauge ﬁelds the integration
over this second Lagrange multiplier will give the boundary condition for the Tdual
coordinate (the “ﬁrst” Lagrange multiplier)49.
After having described the strategy, we will now present the details of the proce1
dure. Setting α = 2 the open string worldsheet action with a nontrivial U (1) gauge
ﬁeld coupling to the boundary reads (for convenience we use a rescaled Aµ as compared
to (2.3.3.2) and choose Minkowskian worldsheet signature here) S= 1
2π d2 z ∂α X µ ∂ α X µ +
M dt ( Aµ ∂t X µ + Vµ ∂n X µ ) , (2.3.3.32) ∂M where M denotes the worldsheet and ∂ M its boundary (parameterized by t). With
∂n we denote the derivative into the direction normal to the boundary. We specify the
48 We will comment brieﬂy on the case of multiple Dbranes at the end of this section.
As we will see below integrating over the second Lagrange multiplier κ boils down to setting an
argument of a delta function to zero. This in turn implies boundary conditions on the Lagrange
multiplier λ.
49 2. Dbrane actions 88 character of the boundary conditions in X 9 direction by the following assignments50
Boundary Condition δX 9 ∂n δX 9 Dirichlet ﬁxed free Neumann free ﬁxed . (2.3.3.33) This implies that for Dirichlet boundary conditions we set A9 = 0 whereas for Neumann boundary conditions V9 = 0 is chosen. For the Neumann boundary conditions
(free varying ends) the variation of S gives the boundary condition (we denote the
normal vector with nα and the tangent vector with tα ) where Fµν 1
nα ∂α X 9 = − F 9ν ∂t xν ,
(2.3.3.34)
2
is the ﬁeld strength of the U (1) gauge ﬁeld Aµ . For Dirichlet conditions we obtain
V9 = 0. (2.3.3.35) Since this should be in agreement with our assignment that the variation of the end
of the open string in the ninth direction is ﬁxed (possibly related to the variations in
other directions), the function Vµ should be interpreted as a vector which is tangent to
the brane. Equation (2.3.3.35) then means that the Dbrane is localized in the ninth
direction.
Since we have chosen the simpliﬁed case of trivial closed string backgrounds any
direction (in cartesian target space coordinates) is an isometry. Suppose that in addition the x9 derivative of the U (1) gauge ﬁeld is pure gauge, i.e. zero modulo gauge
transformations. So, without loss of generality we restrict ourselves to the case that
the gauge background is X 9 independent. We also assume that the tangent vector Vµ
does not depend on X 9. We specify the boundary condition on X 9 by the equation
bα∂α X9 M = independent of X 9,
∂ (2.3.3.36) where bα is a worldsheet vector with a given orientation to the boundary. In case
of Dirichlet boundary conditions, bα is parallel to the boundary (bα = tα ). For free
varying ends (Neumann boundary conditions) bα is normal to the boundary. In the
action (2.3.3.32) X 9 does not mix with the other ﬁelds. We focus on the X 9 dependent
part 50 ¯
S = S + S (9),
1
d2 z ∂α X 9 ∂ α X 9 +
S (9) =
2π
M (2.3.3.37)
dt A9 ∂t X 9 + V9∂n X 9
∂M ,(2.3.3.38) A ﬁxed boundary condition on a variation means that this variation depends on the boundary
values of variations of other ﬁelds (or is zero). In particular, if we do not vary the other directions we
can replace “ﬁxed” by “zero” in (2.3.3.33). 2. Dbrane actions 89 ¯
where S stands for the X 9 independent part. The action is invariant under constant
shifts in X 9. We transform this into a local symmetry by the replacement
∂αX 9 → Dα X 9 = ∂α X 9 + Ωα , (2.3.3.39) where Ωα is the isometry gauge ﬁeld. (We use this terminology in order to avoid
confusion with the open string excitation mode Aµ .) The isometry gauge ﬁeld Ωα
transforms under local shifts in X 9 such that DαX 9 is invariant. We introduce a bulk
Lagrange multiplier λ in order to constrain the Ωﬁeld strength51
f= αβ ∂α Ωβ (2.3.3.40) to vanish. Further, we add a second boundary Lagrange multiplier κ whose task is to
ﬁx the boundary condition of Ωα . Taking into account the Lagrange multipliers, the
gauged action reads52
(9) Sgauged = 1
2π M 1
2π
1
+
2π d2z ∂α X 9∂ αX 9 + ΩαΩα + 2Ωα ∂ αX 9 − 2 + ∂M
∂M αβ Ω β ∂α λ dt A9∂t X 9 + V9∂n X 9
dt (A9tα + V9nα + κbα + 2λtα) Ωα , (2.3.3.41) where for later convenience we have performed partial integrations such that no derivative of Ωα appears in the action. The worldsheet vector tα denotes the tangent vector
to the boundary. The Tdual model will be obtained by integrating out Ωα . The
Tdual coordinate will be λ. Its boundary conditions are going to be ﬁxed by the
integration over κ. Before going through the steps of this prescription, let us verify
that the gauged action is equivalent to the ungauged one. Integration over λ leads to
Ωα = ∂α ρ, (2.3.3.42) where ρ is an arbitrary worldsheet scalar. Integrating out κ leads to the boundary
condition
bα∂α ρ = 0. (2.3.3.43) Because neither the background (nor bα) depend on X 9, the scalar ρ can be absorbed
completely into a redeﬁnition of X 9 without spoiling the boundary condition (2.3.3.36).
(In addition ρ needs to live on a circle with radius equals the compactiﬁcation radius.
51 In two dimensions we can hodge dualize the two form ﬁeld strength to a scalar f .
As before we write the case of D and N boundary conditions into one formula. Recall that A9 = 0
and bα = tα for D boundary conditions, and V9 = 0 and bα = nα for N boundary conditions.
52 2. Dbrane actions 90 This issue has been addressed in [15]. The discussion given there leads to the observation that λ lives on a circle with inverted radius.) Hence, the gauged and ungauged
models are equivalent.
In order to construct the Tdual model we ﬁrst integrate out Ωα. Because the
action (2.3.3.41) does not contain any derivatives of Ωα (it is ultra local with respect
to the isometry gauge ﬁeld), the functional integral over Ωα factorises into a bulk
integral and a boundary integral
DΩM∪∂ M (. . . ) = DΩM (. . . ) × DΩ∂ M (. . . ) . (2.3.3.44) Integrating out Ω in the bulk leads to the ungauged bulk action with X 9 replaced by
λ. This is exactly as in the closed string computation (up to a boundary term)
˜(9)
Sbulk =
= 1
2π
1
2π M d2z ∂α λ∂ αλ + 2 αβ (2.3.3.45) dt 2λ∂tX 9, d2z ∂α λ∂ αλ +
M α λ∂β X 9 (2.3.3.46) ∂M where in the second line we have used Stokes theorem.
The additional ingredient comes from the second factor in (2.3.3.44). This gives a
two dimensional delta function
DΩ∂ M e−Sgauged,∂ M ∼ δ 2 (A9tα + V9nα + κbα + 2iλtα) . (2.3.3.47) Let us evaluate this delta function for the two cases: X 9 has Dirichlet boundary
conditions (bα = tα ) or Neumann conditions (bα = nα ). In the ﬁrst case, the evaluation
of the delta function ﬁxes κ in terms of λ and sets V9 = 0. This means that λ has free
varying ends, i.e. Neumann boundary conditions. Taking into account the boundary
term in (2.3.3.46) we obtain that the dual U (1) gauge ﬁeld is determined by the
position of the original Dbrane,
˜
Aλ = −2X 9∂ M . (2.3.3.48) Recall that the original Dirichlet boundary condition may depend on the other directions, i.e. the rhs of (2.3.3.48) is some ﬁxed function.
If X 9 satisﬁes Neumann conditions, the evaluation of the delta function leads to
κ = 0 and the Dirichlet boundary condition
1
λ∂ M = − A9 .
2 (2.3.3.49) In the Tdual string theory there is a Dbrane located in x9 along the curve A9 (note
that A9 may depend on the coordinates diﬀerent from x9 ). Note also that plugging 2. Dbrane actions 91 the boundary condition (2.3.3.49) into (2.3.3.46) cancels the original A9 coupling to
the boundary.
To summarize, we have seen that Tduality interchanges Dirichlet with Neumann
boundary conditions. The position of the Dbrane is interchanged with the U (1)
gauge ﬁeld component in the Tdualized directions. Starting with Neumann boundary
conditions it is easy to see that gauge transformations do not change the sigma model
for the string, i.e. the ﬁeld equations of the string excitations do not depend on gauge
transformations. Via Tduality this translates to changes of the position of a Dbrane,
in particular constant shifts are moduli of the theory. From the above expressions it
is also clear that performing the Tduality twice will result in the original theory.
With these considerations we can go back to the eﬀective action (2.3.3.20) and
generalize it to non space ﬁlling branes. This is done by simply replacing the Aµ
components where µ labels a Dirichlet direction by scalars. These scalars are the
collective coordinates of the lower dimensional Dbrane. One can also parameterize
the worldvolume of the Dbrane by an arbitrary set of parameters. In this case one
needs to replace bulk ﬁelds by the induced quantities. The eﬀective Dbrane action
for lower dimensional Dbranes can be also computed in the sigma model approach
directly. This has been done in [315].
Finally, let us comment brieﬂy on the case of multiple branes. We start with
Neumann boundary conditions. The gauge ﬁeld A9 is now a matrix. Suppose that
this matrix is diagonal. In this case the above discussion is valid if we just replace A9
by a diagonal matrix everywhere. In the Tdual theory, the position of the Dbrane is
a diagonal matrix. The interpretation is that each entry corresponds to the position
of a single Dbrane. The matrix describes a set of Dbranes. The more general case of
nondiagonal gauge ﬁelds is rather complicated. It is addressed e.g. in[143, 140, 141].
2.3.3.3 RR ﬁelds So far, we have discussed Dbrane eﬀective actions only for trivial RR backgrounds.
The reason was mainly of technical origin. It is rather complicated to describe nontrivial RR backgrounds in a sigma model approach. Later in section 4.3, we will use
such a description for a particular background. Now, we will not discuss the RR
background in a sigma model. Instead we will use our computation of section 2.3.2
and ﬁeld theoretic arguments.
In section 2.3.2 we have seen that the Dpbrane carries RR charge with respect
to a p + 1 form RR gauge potential of type II theories. In section 2.3.3.1 we argued
that the interaction of Dbranes via closed strings is obtained by adding the eﬀective
Dbrane action to the eﬀective type II action (IIA for even p, and IIB for odd p). 2. Dbrane actions 92 Combining these two observations, we infer that the eﬀective Dbrane action contains
an additional piece
√
3−p
π
4π 2α 2
S1 = SDBI +
dp+1xA1,... ,p+1 ,
(2.3.3.50)
κ
where we assume that the Dbrane worldvolume extends along the ﬁrst p + 1 dimensions. (In general, the Dbrane can be parameterized by a set of p + 1 parameters. In
this case, the Dbrane action is written in terms of induced ﬁelds.) We have abbreviated the action (2.3.3.20) with SDBI . Further, we used the result (2.3.2.31) to ﬁx the
coeﬃcient in front of the RR coupling.
The label in S1 has been introduced because now we will argue that there are
further couplings to RR ﬁelds. These occur if another a Dbrane lies within the
worldvolume of the considered Dbrane, or a Dbrane intersects the considered Dbrane. In such a case there will be strings starting and ending on diﬀerent Dbranes.
They give rise to massless ﬁelds transforming in the fundamental representation of
the gauge group living on the considered Dbrane. Under certain circumstances there
may be chiral fermions leading to potential gauge anomalies. Such anomalies can be
canceled by assigning anomalous gauge transformations to certain bulk RR ﬁelds and
adding an interaction term to the eﬀective Dbrane action. This procedure has been
carried out in detail in[218, 145]. Here, we just brieﬂy give the result.
In cases that there is an anomaly, this anomaly can be canceled by adding a ChernSimons term to the Dbrane action
S = S1 + SCS , (2.3.3.51) with (for N coincident Dbranes – for N > 1 also the DBI action needs to be modiﬁed
as discussed in the end of section 2.3.3.1)
SCS = iF Bp C ∧ tr e 2π ˆ
A (R). (2.3.3.52) The way of writing the ChernSimons term needs explanation. The integral is taken
over the worldvolume of the Dpbrane which is denoted by Bp . The integral is a
formal expression in diﬀerential forms. It is understood that only p + 1 forms out of
this expression are kept.
The ﬁrst form C is an RR q + 1 form where q is the spatial dimension of the surface
in which the two Dbranes (or sets of Dbranes) overlap. The last term contains the
so called Aroof genus. This is a polynomial in the curvature twoform (for an explicit
deﬁnition see e.g. [218]). In addition to adding SCS to the Dbrane action the RR form
C receives a contribution under gauge transformations. This comes about as follows. 2. Dbrane actions 93 The deﬁnition of the RR ﬁeld strength receives a correction (the correction is related
to a ChernSimons form whose explicit form is not needed here)
H = dC + correction (2.3.3.53) such that
iF dH = 2πδ (Bp → M10) TrN e 2π ˆ
A (R), (2.3.3.54) where the delta function means that this correction is supported on the worldvolume
of the Dbrane, only. Even though the right hand side of (2.3.3.54) is gauge invariant,
C has to change under gauge transformations in order to ensure that H is invariant.
The construction is such that the change of SCS under gauge transformations cancels
the anomaly.
2.3.3.4 Noncommutative geometry It is interesting to observe that the Dbrane action can be expressed as a noncommutative gauge theory. Here, noncommutative must not be confused with non Abelian.
It does not refer to the gauge group but to a property of space. Before sketching the
connection to string theory, we will brieﬂy give some basic ingredients of noncommutative ﬁeld theory. In diﬀerence to commutative ﬁeld theory it is assumed that the
coordinates of Rn do not commute (we indicate this by putting a hat on the coordinate)
xi , xj = iθij ,
ˆˆ (2.3.3.55) where we restrict to the case that θij are cnumbers. Because of the non commuting
coordinates one has to specify the ordering in say complex functions. For our purpose
the Weyl ordering is appropriate. The Weyl ordering is constructed as follows. The
starting point is the pair of the function and its Fourier transform in commutative
space (ﬁrst with commuting coordinates)
φ (x) = 1
(2π ) ˜
dn k eikx φ (k) . n
2 (2.3.3.56) The Weyl ordered functions are deﬁned by replacing the commuting coordinates xi
with the non commuting ones xi in (2.3.3.56) (but keeping k as a commutative inteˆ
gration variable),
φW (ˆ) =
x 1
(2π ) n
2 ˆ˜
dn k eikx φ (k) . (2.3.3.57) 2. Dbrane actions 94 A natural prescription to multiply two Weyl ordered functions is
(φW ψW ) (ˆ) ≡ φ (ˆ) ψ (ˆ) =
x
x
x 1
(2π )n x
ˆ˜
˜
dn kdn q e−i(k+q)ˆ eikx φ (q + k) ψ (−k) . (2.3.3.58)
Multiplying the two exponentials on the rhs of (2.3.3.58) using the BCH formula and
afterwards dropping the hat on the coordinates leads to a natural way to deform the
algebra of ordinary functions (φ: commuting Rn → C) by replacing the ordinary
product by the Moyal product
(φ ψ ) (x) = e iθ ij ∂
∂
∂xi ∂y j φ (x) ψ (y )x=y . (2.3.3.59) This deformed algebra is noncommutative but still associative. In the limit θij → 0 it becomes the familiar commuting algebra (ordinary multiplication in C).
Noncommutative ﬁeld theories – as we will meet them on Dbranes– are roughly
obtained as follows. One takes the ordinary action for the ﬁeld theory and replaces products of ﬁelds by the Moyal product (2.3.3.59). (This is only a very rough prescription since for example any “zero” can be expressed as the commutator with respect to
the ordinary product which becomes something nontrivial after the deformation. An
additional principle is for example given by the requirement that the deformed action
should posses the same (but possibly deformed) symmetries as the commutative one.)
Our starting point for connecting Dbranes to noncommutative ﬁeld theory is a
slightly rescaled version of the non linear sigma model (2.3.3.2)53
S= 1
4πα M d2z Gij ∂α X i∂ α X j − 2πiα Bij αβ ∂α X i∂β X j . (2.3.3.60) A possible U (1) gauge background could be absorbed into Bij by use of Stoke’s theorem. However, we will restrict ﬁrst to the case that Bij is constant, and add a U (1)
gauge ﬁeld coupling to the boundary later. Note also that Bij has now mass dimension
two – the canonical dimension of a gauge ﬁeld strength. We consider the case that
all coordinates X i have Neumann boundary conditions (coordinates with Dirichlet
boundary conditions do not play a role here and may be added as spectators). The
propagator for the X i can be easily obtained from the expressions (2.3.3.11). The
53 The index i instead of µ indicates here that we focus on space like target space dimensions. The
rescalings of ﬁelds have been done mainly in order to achieve agreement with the literature[418]. 2. Dbrane actions 95 redeﬁned quantities are
ˆ
G−1 ij = 1
G + 2πα B
2 ˆ
Gij = Gij − 2πα
θij ij
S B G−1 B 1
G + 2πα B = 2πα = − 2πα 1
1
G
G + 2πα B G − 2πα B = ij , , (2.3.3.61)
(2.3.3.62) ij
A 1
1
B
G + 2πα B G − 2πα B 2 ij ij . (2.3.3.63) In particular the open string ends propagate according to (call z 1 = τ )
X i (τ ) X j τ ˆ
G−1 = −α ij log τ − τ 2 i
+ θij
2 τ −τ , (2.3.3.64) where the epsilon function is equal to the sign of its argument, and zero for vanishing
argument.
Let us pause for a moment and explain how the last term in (2.3.3.64) arises. The
propagator (2.3.3.11) contains a term ( a factor of α appears now in the deﬁnition of
θij (2.3.3.63))
− z −z
¯
θij
.
log
2π
z−z
¯ (2.3.3.65) We take z = τ + iσ (hoping that this does not cause confusion due to the fact that
now τ and σ parameterize the upper half plane whereas they parameterized a strip
earlier (and will so in later sections)). Ordering with respect to real and imaginary
part, one obtains for (2.3.3.65)
− θij
log
2π (τ − τ )2 + 2i (σ + σ ) (τ − τ )
(τ − τ )2 + (σ + σ )2 . (2.3.3.66) Using the relation
log z = log z  + i arg (z )
and taking the limit σ + σ → +0 one obtains
i
− θij 1 −
2 τ −τ . (2.3.3.67) Dropping an irrelevant constant, this yields the last term in (2.3.3.64).
In the following we will be interested in the α → 0 limit (while keeping θij ﬁxed),
where the propagator (2.3.3.64) takes the form
X i (τ ) X j (0) = i ij
θ (τ ) .
2 (2.3.3.68) 2. Dbrane actions 96 With this propagator one can compute the following operator product
i i ij
p i : eipi x (τ ) :: eiqi x (0) : = e− 2 θ i qj (τ ) : eipi X i (τ )+iq iX i (0) :, (2.3.3.69) where the normal ordering means that self contractions within the exponentials are
subtracted. By use of Fourier transformation one can deduce the operator product for
generic functions
i ij
θ : φ (X (τ )) :: ψ (X (0)) : = : e 2 ∂2
∂X i (τ )∂X j (0) φ (X (τ )) ψ (X (0)) : . (2.3.3.70) In the limit of coincident arguments the operator product can be related to the Moyal
product
lim : φ (X (τ )) ψ (X (0)) : = (φ ψ ) (X (0)) . (2.3.3.71) τ →+0 This expression suggests that we are likely to obtain noncommutative ﬁeld theory if
we use the limiting procedure on the lhs of (2.3.3.71) as a way to regularize composite
operators. This regularization technique is known as point splitting. In composite
operators well deﬁned (normal ordered) parts are taken at diﬀerent points, and then
the limit to coinciding points is performed (after adding counterterms if needed).
In the following we are going to argue that we obtain an eﬀective noncommutative
theory on the Dbrane if we use the point splitting regularization instead of dimensional
(or PauliVillars) regularization. For a trivial worldsheet metric point splitting simply
means that we cut oﬀ short distances by keeping
τ − τ > δ, (2.3.3.72) and take δ to zero in the end. First, we add the following interaction term to (2.3.3.60)
Sint = −i dτ Ai (X ) ∂τ X i. (2.3.3.73) Classically this term is invariant under a gauge transformation
δAi = ∂i λ. (2.3.3.74) Now, we are going to observe that whether or not the partition function is invariant
depends on the regularization prescription. To this end, note that δZ contains a term
δZ = − dτ Ai (X ) ∂τ X i · dτ ∂τ λ + . . . . (2.3.3.75) Schematically this integral has the form
dτ dτ ∂τ f X i (τ ) X j τ = dτ f X i (τ ) X j τ τ =τ −δ
τ =τ +δ . (2.3.3.76) 2. Dbrane actions 97 If we treat the divergence at τ = τ with dimensional regularization (as we did in
section 2.3.3.1) this expression vanishes since it does not matter from which side we
approach the singularity. (The epsilon function in the propagator is zero at τ = τ
and the logarithms are replaced by the regularized expressions.)
If, however, we choose the point splitting method (2.3.3.72) instead, we obtain
δZ = −
=− dτ : Ai (X (τ )) ∂τ X i (τ ) :: λ (X (τ − 0)) − λ (X (τ + 0))
dτ : (Ai λ − λ Ai ) ∂τ X i : + . . . , (2.3.3.77) where in the second step the connection between the operator product and the Moyal
product (2.3.3.71) has been used. Hence, when using the pointsplitting regularization
(2.3.3.72), the string partition function is not invariant under ordinary gauge transformations. However, the lack of invariance can be cured by replacing the gauge ﬁeld
ˆ
Ai with a “noncommutative” gauge ﬁeld Ai with the deformed gauge transformation
ˆˆ
ˆ
ˆ
δAi = ∂i λ + iλ Ai − iAi λ. (2.3.3.78) Such a transformation is a gauge symmetry in the noncommutative version of (U (1))
YangMills theory. The gauge invariant ﬁeld strength is
ˆ
ˆˆ
ˆ
ˆ
ˆ
Fij = ∂i Aj − ∂j Ai − iAi Aj + iAj ˆ
Ai . (2.3.3.79) Indeed, computing the eﬀective action of the open string with the pointsplitting
method, one ﬁnds the noncommutative version of the DiracBornInfeld action (2.3.3.20).
We will not go through the details here, but refer the interested reader to[418] and
further references to be given in the end of this review.
The eﬀective Dbrane action was obtained by setting open string beta functions
to zero. Now, we have seen that the outcome can depend on the way we regularize
singularities: commutative DiracBornInfeld e.g. for dimensional regularization and
noncommutative DiracBornInfeld for pointsplitting. From quantum ﬁeld theory it
is known that beta functions which diﬀer by the way of renormalization should be
identical up to redeﬁnitions of the couplings. In our example the couplings are Ai in
ˆ
the commutative case, and Ai in the noncommutative one. Therefore, there should
exist a ﬁeld redeﬁnition relating commutative gauge theory to noncommutative one.
Indeed, such a ﬁeld redeﬁnition has been found in[418], it is sometimes called the
SeibergWitten map.
The connection between Dbranes and noncommutative ﬁeld theory has many interesting aspects, which we will, however not further discuss in this review. 2. Orientifold ﬁxed planes 2.4 98 Orientifold ﬁxed planes In this section we will introduce an extended object which is called orientifold ﬁxed
plane. This is nothing but the orbifold plane of section 2.2 whenever the corresponding
discrete target space mapping is combined with a worldsheet parity inversion. (Recall
that an orbifold ﬁxed plane was deﬁned as an object being invariant under an element
of a discrete group acting on the target space.)
At ﬁrst we will study unoriented closed (type II) strings. These are closed strings
which can be emitted or absorbed by an orientifold ﬁxed plane. Afterwards we will
investigate how orientifold ﬁxed planes interact via closed strings. We will learn that
orientifold ﬁxed planes carry tension and RR charges. In particular, RR charge conservation implies that orientifold ﬁxed planes cannot exist whenever they possess compact
transverse dimensions. However, by adding Dbranes one can construct models containing orientifold planes with transverse compact dimensions. Such constructions
are known as orientifold compactiﬁcations. We will present the type I theory and an
orientifold analogon of the K3 orbifold discussed in section 2.2.2. (Type I theory is
actually not a compactiﬁcation. Here, the orientifold planes are space ﬁlling and do
not have transverse dimensions. However, the construction falls into the same category
as orientifold compactiﬁcations.) 2.4.1 Unoriented closed strings Recall the mode expansions for type II strings (now with 0 ≤ σ < 2π ). The general
solution to the equation of motion for the bosons is
µ
µ
X µ = XR σ − + XL σ + , (2.4.1.1) with
µ
XR =
µ
XL = 1µ 1µ− i
x + pσ +
2
2
2
1µ 1µ+ i
x + pσ +
2
2
2 n=0 n=0 1 µ −inσ−
αe
,
nn (2.4.1.2) 1 µ −inσ+
αe
˜
.
nn (2.4.1.3) The mode expansions for the worldsheet fermions are
µ
ψ− =
µ
ψ+ − dµ e−inσ ,
n
n∈Z =
n∈Z (2.4.1.4) +
˜
dµ e−inσ ,
n (2.4.1.5) 2. Unoriented closed strings 99 in the R sectors, and
− µ
ψ− = bµe−irσ ,
r (2.4.1.6) ˜µe−irσ+
br (2.4.1.7) r ∈Z+ 1
2 µ
ψ+ =
1
r ∈Z+ 2 in the NS sectors.
We deﬁne an operator Ω which changes the orientation of the worldsheet. For the
closed string the action of Ω is
Ω : σ ↔ −σ. (2.4.1.8) For left handed fermionic modes, we introduce an additional sign such that the product
of a left with a right handed fermionic mode is Ω invariant (recall that fermionic modes
from the left moving sector anticommute with fermionic modes from the right moving
sector). In formulæ, this means
Ωαµ Ω−1 = αµ , Ωbµ Ω−1 = ˜µ, Ω˜µ Ω−1 = −bµ ,
˜n
br
br
n
r
r
µ −1
˜µ , Ωdn Ω−1 = −dµ .
˜
Ωdn Ω = dn
n (2.4.1.9) From this we see that Ω is a symmetry in type IIB theory – the only closed superstring
which is leftright symmetric. (Note that the GSO projection operator (2.1.2.59) in
the R sector contains an even number of d0 ’s. Hence, the sign in the transformation
+
˜+
(2.4.1.9) cancels out and e.g. PGSO is interchanged with PGSO .)
Let us study the action of Ω on the massless sector of type IIB excitations. We
take the vacuum to be invariant under worldsheet parity reversal. The massless NSNS
sector states are (in light cone gauge)
b
bi 1 ˜j 1 k
−−
2 (2.4.1.10) 2 The action of Ω on this state interchanges the indices i and j . Thus the states surviving an Ω projection are symmetric in i, j – these are the graviton Gij and the dilaton
Φ. Since Ω relates the NSR with the RNS sector only invariant superpositions are
kept. Thus we obtain only one gravitino (56 components) and one dilatino (8 components). Half of the target space supersymmetry is broken by the Ω projection. The
massless states in the RR sector are obtained from the tensor product of the left with
the right moving R vacuum. The R vacua are target space spinor components and
Ω interchanges the left with the right moving vacuum. Because spinor components
anticommute the antisymmetrized tensor product survives the Ω projection. This is
the 28 dimensional SO(8) representation – the antisymmetric tensor Bij . We obtain 2. Unoriented closed strings 100 the ﬁeld content of the heterotic string without the internal fermions λA . As we stated
+
before, a theory with such a massless spectrum suﬀers from gravitational anomalies. In
the heterotic theory we actually needed 32 worldsheet fermions λA whose quantization
+
provided exactly the gauge multiplets needed to obtain an anomaly free massless spectrum. Later we will see that one needs to add D9branes to the unoriented type IIB
theory, for consistency. Before going into that let us study for a while the unoriented
closed stringtheory – even though it is not consistent yet.
The theory of unoriented type IIB strings contains orientifoldnineplanes – or short
O9planes. An Oplane is a set of target space points which is ﬁxed under an element
of a discrete group which contains Ω (the element must contain Ω). Because Ω alone
does not act on the target space geometry the full target space is ﬁxed under Ω. The
ﬁxed set of points is space ﬁlling – it is an O9plane.
We have seen that when we compactify the type IIB string on a circle and perform
a Tduality we obtain type IIA theory compactiﬁed on a circle with inverted radius.
Let us study what happens to the O9planes in this process. Formally, we have the
expression (X 9 stands for the bosonic string coordinate)
ΩX 9Ω−1 TDUALITY −→ T ΩT −1 T X 9T −1 T ΩT −1 −1 . (2.4.1.11) We want to know the Tdual of Ω which is denoted by T ΩT −1 . This can be computed
as follows. We ﬁrst perform a Tduality, then act with Ω on the Tdual coordinate,
and ﬁnally Tdualize back. These steps are collected in the following diagram (we use
(2.1.5.31) for Tduality)
T Ω T −1 9
9
9
9
9
9
9
9
XL + XR −→ XL − XR −→ XR − XL −→ −XR − XL (2.4.1.12) Thus we see that T ΩT −1 reﬂects the dimension in which T acts, and also interchanges
left with right movers (the second statement can be easily veriﬁed by drawing the
diagram (2.4.1.12) for the left or right moving piece alone). Thus, for Tduality in X 9
direction we can write
T ΩT −1 = R9Ω, (2.4.1.13) R9 : X 9 → −X 9. (2.4.1.14) where R9 is the Z2 element The action on the worldsheet fermions can be studied likewise. Now we go to the
decompactiﬁcation limit on the type IIA side. Instead of an O9plane we have an
O8plane, because now only points with X 9 = 0 are ﬁxed under the action of ΩR9.
Repeating this argumentations for more than one Tdualized circle we conclude that 2. Oplane interactions 101 1
+2 1
2 = Figure 2.12: The superposition of two strings with opposite orientation can be viewed
as a crosscap. The crosscap is a circle with diagonally opposite points being identiﬁed.
we have Opplanes with even (odd) p in type IIA (B) theory. For an Opplane with
even p, Ω comes combined with a Z2 operator reﬂecting an odd number of dimensions.
+
˜−
In particular, this combination interchanges e.g. PGSO with PGSO , i.e. it is indeed a
symmetry of type IIA strings. The closed string is unoriented only when it is located
on an Oplane. A string oﬀ the Oplane is oriented. Its counterpart with the opposite
orientation is the R9 image of the string. 2.4.2 Oplane interactions An Oplane is deﬁned as an object where closed strings become unoriented when they
hit it. Topologically this can be depicted by a crosscap as illustrated in ﬁgure 2.12.
The opposite process is a crosscap decaying into a pair of strings with diﬀerent
orientations. Only one string out of this pair is physical – the other one is the ΩR
image, where R now stands for a target space mapping leaving the Oplane ﬁxed. Thus
Oplanes can emit or absorb oriented strings. They possibly interact via the exchange
of closed oriented strings. This indicates that there is an interaction among Oplanes
and between Dbranes and Oplanes. We are going to study these interactions in the
following two subsections.
2.4.2.1 Oplane/Oplane interaction, or the Klein bottle In ﬁgure 2.13 we have drawn a process in which two Oplanes interact via the exchange
of closed strings. We restrict to the special case that the orientifold group element ΩR
squares to one. (Combining orbifold compactiﬁcations with orientifolds, one can have
the more general situation that the orientifold group elements square to a nontrivial
orbifold group element. This has to be the same for the two Oplanes. Then a twisted
sector closed string is exchanged.)
In the following we are going to compute this process. As in the Dbrane computation, we take the Oplanes to be parallel. The range for the worldsheet coordinates
is
0 ≤ σ < 2π , 0 ≤ τ < 2πl. (2.4.2.1) 2. Oplane interactions 102 τ
σ Oi Oj Figure 2.13: Oplane Oi and Oplane Oj talking to each other by exchanging a closed
string. Figure 2.14: The triangulated version of ﬁg. 2.13 on the left. By manipulations
preserving the topology this can be mapped onto the triangulated version of a Klein
bottle on the right.
Like in section 2.3.2 we want to perform the computation in the scheme where the
role of τ and σ are interchanged – i.e. in the worldsheet dual channel. In this dual
channel, a virtual pair of closed strings pops out of the vacuum – one of the strings
changes its orientation before they rejoin. Therefore, this is called the loop channel.
Before performing the transformation to the loop channel, we need to describe the tree
channel process ﬁg. 2.13 such that it is periodic in time. The method of doing so diﬀers
slightly from the Dbrane/Dbrane interaction. It is best explained by looking at the
triangulated version of picture 2.13 and its double cover which is a torus. We draw this
in ﬁgure 2.14. In the left picture, the shaded region is the triangulated version of ﬁgure
2.13. The halfcircles indicate the identiﬁcations of the crosscaps. The white region
shows our intention to obtain a description which is periodic in τ , with a period 4πl.
Now, one cuts the shaded part along the dotted line (with the indicated orientation),
and ﬂips the upper rectangle once around its right vertical edge and afterwards shifts
it down in the vertical direction. We obtain a process which is indeed periodic in τ ,
and now τ ∈ [0, 4πl). (This periodicity appears due to the crosscap identiﬁcations 2. Oplane interactions 103 indicated by the left half circles. The crosscap identiﬁcations for the other Oplane
ensure that one can glue the upper rectangle to the lower one after the ﬂip and the
shift.) It is diﬃcult to describe this process as a tree channel closed string exchange.
Instead we can interchange the roles of σ and τ . Then the interpretation is that a pair
of closed strings of length 4πl pops out of the vacuum, one of the closed strings changes
its orientation before they annihilate after a worldsheet time π . This is a vacuum loop
amplitude which has the topology of the Klein bottle. The parameter range is
0 ≤ σ < 4πl , 0 ≤ τ < π. (2.4.2.2) As in the Dbrane case (section 2.3.2) we want to rescale the dualized worldsheet
coordinates such that their ranges are the canonical ones, which (now for the closed
string) are
0 ≤ σ < 2π , 0 ≤ τ < 2πt. (2.4.2.3) This can be achieved by the redeﬁnitions
τ → τ 2t , σ → σ
.
2l (2.4.2.4) For the Hamiltonian this induces a rescaling
H → 2l (2t)2 H. (2.4.2.5) Analogous to the annulus discussion in section 2.3.2 we require that the action of the
rescaling on the time evolution operator is
e−πH → e−2πtH . (2.4.2.6) This yields a relation between l and t
1
lt = .
4 (2.4.2.7) Periodic boundary conditions on fermions along the vertical axis of the lhs of ﬁgure
˜
2.14 correspond to a (−)F = (−)F insertion whenever the vertical axis is identiﬁed with the worldsheet time on the rhs of ﬁgure 2.14. Only closed strings for which the
rightmoving (−)F eigenvalue equals the leftmoving one contribute to the Klein bottle
amplitude. (In the R sector an additional sign may occur depending on whether we
are looking at type IIA or IIB strings. This does not matter here since the Rsector
contributions with a (−)F insertion vanish anyway.)
Since the connection between tree level periodicities and loop channel insertions is
a bit less obvious than in the Dbrane/Dbrane interaction, let us explain it in some 2. Oplane interactions 104 detail here. We take the parameter range (2.4.2.3). We are interested in the behavior
of worldsheet fermions under shifts in τ by 4πt. Periodic behavior corresponds to tree
level RR exchange whereas antiperiodicity translates to NSNS exchange. Fig. 2.14
tells us how to continue in τ beyond 2πt
˜ µ
µ
ψ± (τ + 4πt, σ ) = (−)F +F ψ± (τ + 2πt, 2π − σ ) , (2.4.2.8) ˜ where the (−)F +F reﬂects the boundary condition on worldsheet fermions under 2πt
˜ shifts in τ . However, in the Klein bottle amplitude only states with (−)F = (−)F
contribute because of an Ω insertion in the trace over states. Therefore, the additional
factor in (2.4.2.8) is not relevant. Now, the 2πt shift in τ can be replaced by acting
with the trace insertion (−)F Ω. The (−)F insertion just cancels a sign included in
the deﬁnition of the trace over fermions for the right movers. By the same token we
˜
have to insert a (−)F for the left movers. Thus we obtain
˜ µ
µ
µ
ψ± (τ + 4πt, σ ) = (−)F Ωψ± (τ, 2π − σ ) Ω−1 = ψ± (τ, σ ) , (2.4.2.9) where in the last step we used our deﬁnition for Ω (2.4.1.9) and the 2π periodicity in
σ . Thus the (−)F insertion in the loop channel ﬁlters out the RR tree level exchange, indeed. Strictly speaking the above consideration is correct only when the fermions
point in directions longitudinal to the Oplane (where the Z2 reﬂection R acts as the
identity). For the other directions there are two signs canceling each other and leading
to the same result. At ﬁrst, there is an additional minus sign in (2.4.2.8) because the
halfcircles in ﬁg. 2.14 now contain also the (nontrivial) action of the Z2 reﬂection R.
This sign is canceled when we replace Ω by ΩR in (2.4.2.9).
We want to ﬁlter out the contribution due to RR exchange in the tree channel.
Then, the loop channel vacuum amplitude is given by the following expression
(−)F −2πα tH
dt
Str ΩR
e
2t
2 = dt
˜
tr ZERO ΩRe−2πα tH0 trBOSONS ΩRe−2πt(HB +HB )
2t MODES
tr NSNS
FERMIONS ΩR (−)F −2πt(HN S +HN S )
˜
e
2 . (2.4.2.10) ˜
Here, we split the Hamiltonian into right and left moving parts H + H and these in
turn into
α H = α H0 + HB + HN S (2.4.2.11) 2. Oplane interactions 105 with54
H0 = p2
4 (2.4.2.12)
∞ 8 HB =
i=1
8 HN S =
i=1 n=1 ∞ r= 1
2 αi n αi −
−
n 1
24 rbi r bi −
−r 1
,
48 , (2.4.2.13) (2.4.2.14) and the corresponding expressions for the left moving sector. The ΩR insertion
projects out contributions of states with zero mode momenta perpendicular to the
Oplanes, since those states are mapped onto states with the negative momentum
in the perpendicular direction by the ΩR insertion. The result for the zero mode
contribution reads
tr ZERO
MODES = 2Vp+1 p+1
dp+1 k −2πα t k2
2
2 = 2·2 2 V
p+1 8α π t
p+1 e
(2π ) − p+1
2 . (2.4.2.15) Also here there is an additional factor of two, due to the possible orientations of the
closed string. (The trace is taken over oriented strings with an Ω insertion. Another
point of view would be that one needs to add to the picture on the rhs of ﬁgure 2.14 a
picture with reversed orientations on the horizontal edges.) For the traces over excited
states we note that the insertion ΩR in the trace means that only states contribute
which are eigenstates of ΩR. This means that the left moving excitations have to be
identical to the right moving ones. Thus, it is straightforward to modify the calculation
presented in section 2.3.2 by just changing the power of the arguments in the functions
2.3.2.20 by two (since the identical left and right moving contributions add). We obtain
1 ˜ trBOSONS ΩRe−2πt(HB +HB ) = 8
f1 (e−2πt ) (2.4.2.16) for the trace over the bosonic excitations, and
tr NSNS
FERMIONS ΩR (−)F −2πt(HN S +HN S )
˜
e
2 18
= − f4 e−2πt .
2 (2.4.2.17) 8
f4 e−2πt
.
8
f1 (e−2πt ) (2.4.2.18) Thus we obtain
dt
(−)F −2πtH
Str ΩR
e
2t
2
1 p+1
− 2 2 Vp+1
2
54 = dt
8α π 2t
2t − p+1
2 From our treatment in section 2.1.2.2 we would get a factor of α 8/2 = 2 instead of α 4/2 = 1 in
the oscillator contributions. Recall, however, that we have changed the σ range from [0, π) to [0, 2π),
˜
meanwhile. We have distributed the zero mode contribution symmetrically on H and H . Taking into
account the eﬀect of rescaling, this gives the factor of 1/4. 2. Oplane interactions 106 Now we undo the worldsheet duality by expressing t in terms of l (2.4.2.7). We use
the transformation properties (2.3.2.22) and take the limit l → ∞ in which the con tribution of massless closed string excitations dominates. This leads to the expression
− 1
2 l→∞ dl 2p+1 Vp+1 4α π 2 − p+1
2 l p−9
2 . (2.4.2.19) We see that the result has almost the same structure as the one we obtained for the
Dbrane/Dbrane interaction in table 2.7. (Recall that now, we separated out the RR
sector exchange.) The diﬀerences are that we do not have the exponential dependence
on the distance and that we do have an additional factor of 2p+1. The explanation for
the missing exponent is very simple. Since the orientifold planes are all located at a
ﬁxed point of the Z2 action R, they cannot be separated in target space. (However,
we could for example compactify the dimensions transverse to the brane. In that case
winding modes would play the role of the distance.)
Before we can deduce the ratio of the Oplane RR to the Dbrane RR charge, we
need to discuss a subtlety appearing because we have modded out reﬂections in the
transverse directions. This has the eﬀect that each transverse direction is “half as
long” as in the Dbrane computation. The implications of this eﬀect are best seen
in a ﬁeld theory consideration. The ﬁeld theory result gives a “Coulomb potential”
which is of the structure chargesquared times density. (The density appears as the
inverse of a second order diﬀerential operator.) The charge is obtained as an integral
over the transverse space (analogous to Q = d3xj 0 in electrodynamics). In the
Oplane case this gives a factor of a half per transverse direction as compared to the
Dbrane/Dbrane interaction. On the other hand the density is multiplied by a factor
of two per transverse direction. Hence, the overall neteﬀect of this transformations is
an additional factor of 2p−9 which we need to put by hand into the Oplane/Oplane
result, before we can compare it with the Dbrane/Dbrane calculation.55 Taking this
into account, the ratio of the Dpbrane RR charge µp to the Opplane RR charge µp
µp = 2p−4 µp . (2.4.2.20) We cannot ﬁx the sign by the present calculation since the charges enter quadratically
the expressions we derived so far. Computing also the contributions without the (−)F
insertions to the Klein bottle, one obtains the square of the Oplane tension. Here, we
infer the result by supersymmetry instead. Since the ΩR projection leaves half of the
55 If we performed a detailed ﬁeld theory calculation we would ﬁnd this factor due to the diﬀerent
target spaces (as argued in the text). Later, we will compactify the transverse dimensions. Then this
factor will appear “automatically” due to a Poisson resummation of the sum over the winding modes.
This must be the case since in the compactiﬁed theory Dbranes and OPlanes will have the same
transverse space. 2. Oplane interactions 107 τ
σ Di Oj Figure 2.15: Dbrane Di and Oplane Oj talking to each other via the exchange of
closed strings.
supersymmetries unbroken, the total one loop amplitude should vanish. This tells us
that the ratio of the Dbrane tension Tp to the Oplane tension Tp is
Tp = 2p−4 Tp , (2.4.2.21) where at the present stage of the calculation the sign is not known. In order to ﬁx the
signs in (2.4.2.20) and (2.4.2.21) we need to study the interaction between Dbranes
and Oplanes. We will do so in the next subsection.
2.4.2.2 Dbrane/Oplane interaction, or the M¨bius strip
o So far, we have seen that Dbranes as well as Oplanes interact via the exchange of
closed type II strings. This suggests that also Dbranes interact with Oplanes. Such
a process is depicted in ﬁgure 2.15. We consider the case of parallel Dbranes and Oplanes. This implies that the Dbrane is located in directions where the Z2 reﬂection
acts with a sign and extended along the other directions.
Again, the range for the worldsheet coordinates is
0 ≤ σ < 2π , 0 ≤ τ < 2πl. (2.4.2.22) In order to understand how to perform the worldsheet duality transformation it is
useful to study the triangulated version of the diagram 2.15. The result of this investigation is drawn in ﬁgure 2.16. The right picture is obtained by cutting the left
one along the dashed line ﬂipping the upper rectangular around its right edge and
afterwards shifting it down. Looking at the left picture with time passing along the
vertical axis we see a process in which a pair of open strings pops out of the vacuum.
Both ends of the strings are in the worldvolume of the brane Di . As time goes by one 2. Oplane interactions 108 Di
Di Di Figure 2.16: The triangulated version of ﬁg. 2.15 on the left. By manipulations preserving the topology this is mapped onto the triangulated version of a M¨bius strip
o
on the right.
of the open strings changes its orientation before they ﬁnally annihilate. The topology
of this diagram is called M¨bius strip (or M¨bius band).
o
o
The range for the worldsheet coordinates after interchanging the role of time and
space is
0 ≤ σ < 4πl , 0 ≤ τ < π, (2.4.2.23) whereas the canonical range for the open string parameters is
0 ≤ σ < π , 0 ≤ τ < 2πt. (2.4.2.24) Hence, we perform the rescaling
τ → τ 2t , σ → σ
.
4l (2.4.2.25) For the Hamiltonian this induces
H → 4l (2t)2 H. (2.4.2.26) Finally the time evolution operator should take its canonical form
! e−πH → e−2πtH . (2.4.2.27) 1
lt = .
8 (2.4.2.28) This tells us how to relate l and t Now, we would like to identify which of the loop channel contributions corresponds
to an RR exchange in the tree channel. Periodicity under 4πt shifts in τ translates to
RR tree level exchange and antiperiodicity to NSNS tree level exchange. We use ﬁg.
2.15 to identify
µ
µ
ψ± (τ + 4πt, σ ) = (−)F ψ± (τ + 2πt, π − σ ) , (2.4.2.29) 2. Oplane interactions 109 where the factor of (−)F appears due to the antiperiodic boundary condition of worldsheet fermions under shifts of 2πt in τ . Let us study the case where we insert in the loop
channel trace just Ω (possibly combined with some target space Z2 action which we
will discuss below). Taking into account the sign when a trace is taken over worldsheet
fermions we obtain
µ
µ
ψ± (τ + 4πt, σ ) = Ωψ± (τ, π − σ ) Ω−1 = Ωψ µ (τ, σ − π ) Ω−1 , (2.4.2.30) where in the second step we have used the mode expansions (2.3.1.22)–(2.3.1.29). In
the open string sector we deﬁne the action of Ω as taking σ → π − σ . This ﬁnally
results in µ µ ψ± (τ + 4πt, σ ) = ψ (τ, 2π − σ ) µ
= ψ± (τ, σ − 2π ) , (2.4.2.31) where once again the mode expansion has been used. We deduce that open string R
sector contributions correspond to closed string RR exchange. (This can also easily
be seen in the mode expansions (2.3.1.22)–(2.3.1.29)). The above consideration is
correct only in NN directions (in directions in which the Dbrane extends). For DD
directions there are a couple of signs which cancel each other such that one gets the
same result. Since the Z2 reﬂection R acts with a sign in those directions, the ﬁrst
line in (2.4.2.30) receives an additional minus sign. Looking at the mode expansion
(2.3.1.22)–(2.3.1.29) in DD directions we observe that this sign is canceled when going
to the second line in (2.4.2.30). Because now we need to replace Ω by ΩR, the ﬁrst
line in (2.4.2.31) contains an additional minus sign which again is canceled by using
the DD mode expansion when going to the second line in (2.4.2.31).
In the above consideration we have only speciﬁed how Ω acts on the oscillators,
and not how it acts on the vacuum. (In the closed string we tacitly took the vacuum as
being invariant under Ω leaving in the NSNS sector the metric invariant and projecting
out the B ﬁeld.) The computations of the Dbrane/Dbrane and the Oplane/Oplane
interactions provided the absolute values of the corresponding RR charges. The result
for the Dbrane/Oplane calculation will give the product of the Oplane times the
Dbrane charge. This should be compatible with our previous result. As we will see
in a moment this leaves the two choices that the ΩR eigenvalue of the open string
Rvacuum is ±1. We will choose the minus sign. This corresponds to a Dbrane. The
action on the NS sector can be inferred by supersymmetry, i.e. it should be such that
the complete one loop M¨bius strip amplitude vanishes. The result is that the massless
o
states have eigenvalue minus one. (This holds as well for Neumann directions as for 2. Oplane interactions 110 Dirichlet directions, since a sign due to the diﬀerent mode expansions cancels a sign
due to the nontrivial action of R in Dirichlet directions.)
Now, we have collected all the necessary information needed to write down the
loop channel amplitude which gives the tree channel RR exchange (recall that a (−)F
insertion leads to a vanishing R sector trace)
− 1
dt
trRΩR e−2πα H =
2t
2
dt
tr ZERO ΩRe−2πtα H0 trBOSONS ΩRe−2πtHB
−
2t MODES
1
tr
ΩR e−2πtHR .
R
FERMIONS
2 (2.4.2.32) The expressions for the Hamiltonians can be directly taken from (2.3.2.14)–(2.3.2.18)
with the diﬀerence that we put y = 0 in (2.3.2.15) (because of the ΩR insertion in
the trace only Dbranes at distance zero from the Oplane contribute). With this
diﬀerence the trace over the zero modes gives (see (2.3.2.19)
tr ZERO
MODES = 2Vp+1 dp+1 k −2πtα k2
e
= 2Vp+1 8π 2α t
(2π )p+1 − p+1
2 , (2.4.2.33) From the mode expansion (2.3.1.20), (2.3.1.21) we learn that
ΩR αµ n (ΩR)−1 = (−1)n αµ n .
−
− (2.4.2.34) Modifying the expression (2.3.2.23) accordingly we obtain
1 2 trBOSONS ΩRe−2πtHB = e−iπ 3 8
f1 e i
−π (t+ 2 ) . (2.4.2.35) The next step is to split the product over integers in the deﬁnition of f1 (2.3.2.20) into
a product over even times a product over odd numbers. This gives ﬁnally
trBOSONS ΩRe−2πtHB = 1 .
8
8
f1 (e−2πt ) f3 (e−2πt ) (2.4.2.36) The mode expansion on the fermions (2.3.1.22)–(2.3.1.29) yields
ΩR dµ (ΩR)−1 = eiπn dµ .
n
n (2.4.2.37) Manipulations analogous to the bosonic trace give the result (recall that we have
chosen the ΩR eigenvalue of the R vacuum to be minus one)
−tr R
FERMIONS ΩR −2πtHR
e
2 8
8
= f2 e−2πt f4 e−2πt , (2.4.2.38) 2. Oplane interactions 111 where the 16fold degeneracy of the R vacuum has been taken into account. We are
interested in the contributions due to tree channel RR exchange and have computed
now everything we need to obtain the result. However, in order to specify the action
of ΩR on the NS vacuum one needs to compute the tree channel NSNS exchange. The
requirement that this cancels the RR interaction determines the action of ΩR on the
open string NS vacuum. We leave this as an exercise. The result is that the massless
vector is odd under ΩR. In the computation of the open string NS sector trace it is
useful to apply the identity (2.3.2.21) on the f functions with the shifted arguments
and afterwards to proceed as we did above, i.e. to split the product in the deﬁnitions
of the f ’s into a product over even and over odd numbers.
So far, we obtained the result
− dt
1
ΩRtrR e−2πα H = Vp+1
2l
2 dt
8π 2α t
2t − p+1
2 8
8
f2 e−2pit f4 e−2πt
. (2.4.2.39)
8
8
f1 (e−2πt ) f3 (e−2πt ) Expressing t in terms of l via (2.4.2.28) and using the properties (2.3.2.22) yields ﬁnally
the tree channel infrared asymptotics
1
2 Vp+1
2 dl 4π 2α − p+1
2 2p−4l p−9
2 . (2.4.2.40) l→∞ This expression has to be compared with the second line (RR contribution) of table
2.7 and (2.4.2.19), where (2.4.2.19) has to be multiplied with 2p−9 as discussed earlier.
(For the M¨bius strip we do not need to put such a factor since there is a cancellation
o
between the Oplane charge and the density.) In (2.4.2.40) we have pulled out a factor
of two. If we take the D brane distance y to zero in (2.7) we can write down the
cumulative infrared asymptotics for a system consisting out of one Dbrane and one
Oplane (situated at the origin in the transverse space)
−Vp+1 4π 2α − p+1
2 dll p−9
2 l→∞ 1 − 2p−4 2 . (2.4.2.41) 2 In ﬁeld theory one obtains a result proportional to µp + µp , (recall that µp is the
Dbrane charge and µp the Oplane charge). Thus we obtain ﬁnally the ratio between
Dbrane and Oplane RR charges
µp = −2p−4 µp . (2.4.2.42) The M¨bius strip computation ﬁxed also the sign of this ratio. However, if we assigned
o
an ΩR eigenvalue of +1 to the open string R vacuum we would obtain an additional
minus sign in (2.4.2.42). There is an ambiguity here. In the next section we will use
our results to construct consistent string theories containing Dbranes and Oplanes.
In this construction this ambiguity cancels out. (In some sense it will turn out that our 2. Compactifying the transverse dimensions 112 present choice is the “natural” one.) The ratio of the Dbrane tension to the Oplane
tension can be inferred by supersymmetry
Tp = −2p−4 Tp. 2.4.3 (2.4.2.43) Compactifying the transverse dimensions When we are trying to compactify the transverse directions of a Dbrane and/or an
Oplane we immediately run into problems. These arise as follows. The equation of
motion for the RR ﬁeld under which the Dpbrane (or the Opplane) is charged reads
(for otherwise trivial background)
d Fp+2 = jp+1, (2.4.3.1) where jp is the external U (1) current indicating the presence of the Dbrane (Oplane).
Integrating this equation over a compact transverse space gives zero for the left hand
side and the Dbrane (Oplane) charge on the right hand side. Therefore, the RR
charge on the rhs has to vanish. To overcome this problem one may want to add
Dbranes and Oplanes such that the net RR charge is zero. Since one needs more
than one Dbrane in order to achieve a vanishing net RR charge, one has to specify
how ΩR acts on a set of multiple Dbranes. For example it could (and actually will)
happen that ΩR (anti)symmetrises strings starting and ending on diﬀerent Dbranes.
Technically, we deﬁne a (projective) representation of the Z2 (generated by ΩR) on the
ChanPaton labels carried by open string in case of multiple Dbranes. The generating
element of this representation is denoted by γΩR . The ΩR action on an open string is
ΩR : ψ, ij → (γΩR )ii ΩR (ψ ) , j i −1
γΩR jj . (2.4.3.2) Here, ψ denotes the oscillator content of the string on which ΩR acts in the same way
as discussed previously. In addition, the order of the ChanPaton labels is altered due
to the orientation reversal. Acting twice with ΩR should leave the state invariant.
This leads to the condition
T
γΩR = ±γΩR , (2.4.3.3) i.e. γΩR is either symmetric or antisymmetric. By a choice of basis this gives the two
possibilities
γΩR = I or γΩR = 0 iI −iI 0 . (2.4.3.4) Let N be the number of Dbranes (ΩR images are counted). Then I denotes an N × N
identity matrix for symmetric γΩR and an N × N identity matrix for antisymmetric
2
2 γΩR. 2. Compactifying the transverse dimensions 113 The trace in the open string amplitudes (annulus and cylinder) includes also a
trace over the ChanPaton labels. For the annulus this is simply
N
i,j =1 N ij  ij = δii δjj = N 2 . (2.4.3.5) i,j =1 In the M¨bius strip amplitude the trace over the ChanPaton labels yields the addio
tional factor
N
−1 T
ij  ΩR ij = tr γΩR γΩR = ±N, i,j =1 (2.4.3.6) with the (lower) upper sign for (anti) symmetric γΩR .
2.4.3.1 Type I/type I strings In the following we are going to investigate the case where the compact space is a torus.
The next issue we need to discuss are zero mode contributions due to windings in
the compact transverse dimensions. For open strings windings can appear in Dirichlet
directions. Since ΩR leaves the winding number of a state invariant these contribute to
the annulus, the Klein bottle and the M¨bius strip. Including the sum over the winding
o
numbers into the corresponding traces leads to additional factors. The transverse space
is a 9 − p–torus:
T 9−p = S 1 × · · · × S 1 .
9−p (2.4.3.7) factors For simplicity we take the radii of these S 1 s to be identical and denote them by r. It
is useful to introduce a dimensionless parameter
ρ= r2
α (2.4.3.8) for the size of the compact space.
With this ingredients the sum over the winding modes gives the following factors
dt
(under the 2t integral):
∞ −2πtρw 2 e w =−∞
∞ −πtρw 2 w =−∞ for the annulus, −2πtρw 2 e (2.4.3.9) 9−p e w =−∞
∞ 9−p for the Klein bottle, (2.4.3.10) for the M¨bius strip.
o (2.4.3.11) 9−p 2. Compactifying the transverse dimensions 114 In the annulus we have restricted ourselves to the special case that all Dbranes are
situated at the same point. This conﬁguration gives the correct leading infrared contribution to tree channel amplitude. One can also include distances among the Dbranes
into the computation. In that case the trace over the ChanPaton labels cannot be
directly taken as in (2.4.3.5) because the zero mode contribution depends on the ChanPaton label. Taking the infrared limit in the tree channel removes this dependence
on the ChanPaton labels and gives the same result as our slightly simpliﬁed computation.56 Now, we express t in terms of l using (2.3.2.8), (2.4.2.7) and (2.4.2.28) and
apply the Poisson resummation formula
∞ e−π (n−b)2
a = √
a n=−∞ ∞ e−πas 2 +2πisb . (2.4.3.12) s=−∞ We obtain
l 9−p
2 ρ p−9
2 ∞ − πlw
ρ 2 2 e − 4πlw
ρ 2 e − 4πlw
ρ e for the annulus, l ρ p−9
2 9−p 2 ∞ w =−∞ l 9−p
2 ρ p−9
2 9−p 2 ∞ (2.4.3.13) for the Klein bottle, w =−∞
9−p
2 9−p (2.4.3.14) for the M¨bius strip.
o (2.4.3.15) 9−p 9−p w =−∞ In the IR limit l → ∞ the sums become a factor of one. The common ρ dependent factor is a dimensionless quantity representing the volume of the transverse space
(sometimes denoted by v9−p ). In the Klein bottle as well as in the M¨bius strip
o
9−p . In the M¨bius strip this is simply the number
there is an additional factor of 2
o
of Oplanes. (The number of Rﬁxed points is two per S 1.) Since the Klein bottle
amplitude is proportional to the square of the total Oplane charge we would expect
another factor of 29−p here. However, this is “canceled” by the correction factor we
put earlier in by hand. As promised in footnote 55 this factor appeared automatically
after we compactiﬁed the transverse dimensions. This is good because now we would
miss the argument for putting it in by hand.
Together with our previous results table 2.7, (2.4.2.19) and (2.4.2.40) we obtain
for the infrared limit of the total (tree level RR channel) amplitude
− 1
4α π 2
2 − p+1
2 Vp+1ρ p−9
2 dl N 2 + 322 2N 32 , (2.4.3.16) l→∞ where the sign is correlated with the ± sign in (2.4.3.3). We observe that the contributions of the Dbrane/Dbrane, Oplane/Oplane and Dbrane/Oplane interaction
56 In the limit l → ∞ the distance dependent exponential function in table 2.7 becomes one. 2. Compactifying the transverse dimensions 115 add up to a complete square, proportional to
(32 N )2 . (2.4.3.17) For consistency the total RR charge has to vanish. Thus, we are lead to the conditions
T
γΩR = γΩR (2.4.3.18) N = 32. (2.4.3.19) and Note that the condition (2.4.3.18) is related to our choice that the ΩR eigenvalue of
the R vacuum (in the open string sector) is minus one. Since (2.4.3.18) implies that
we can choose a basis such that
γΩR = I, (2.4.3.20) our choice of the ΩR eigenvalue of the R vacuum seems natural. The case p = 9
can be obtained from our previous considerations in two ways. The simplest is to set
p = 9 in (2.4.3.16). Requiring this expression to vanish yields equations (2.4.3.18)
and (2.4.3.17) independent of p.57 An alternative way is to perform Tdualities with
respect to all compact directions and to take the decompactiﬁcation limit in the Tdual
model. Both methods lead to the same results. The massless closed string spectrum
has been discussed in section 2.4.1. In the NSNS sector one ﬁnds the metric Gij and
the dilaton Φ. The RR sector provides the antisymmetric tensor Bij . The Ω invariant
combinations of the NSR with the RNS sector massless states yield the space time
fermions needed to ﬁll N = 1 supermultiplets.
It remains to study the open string sector. The massless NS sector states are
i
ψ− 1 k, mn λmn . (2.4.3.21) 2 The ΩR image of these states is
i
−ψ− 1 k, mn λT mn . (2.4.3.22) 2 Hence, the ChanPaton matrix λ must be antisymmetric
λ = −λT . (2.4.3.23) The states (2.4.3.21) are vectors and thus should be interpreted as gauge ﬁelds of
a certain gauge group. In order to identify the gauge group, we require that the
57 Recall that a pdependence cancels in the product of Oplane charges with the number of Oplanes. 2. Compactifying the transverse dimensions type I
SO(32) # of Q’s
16 # of ψµ ’s
1 116 massless bosonic spectrum
NSNS
Gµν , Φ
open string Aa in adjoint of SO(32)
µ
RR
Bµν Table 2.8: Consistent string theories in ten dimensions containing open strings.
state (2.4.3.21) transforms in the adjoint and that gauge transformations preserve
the condition (2.4.3.23). Thus the commutator of a generator of the gauge group
with a 32 × 32 antisymmetric matrix (λ) should be a 32 × 32 antisymmetric matrix.
This consideration determines the gauge group to be SO(32). The R sector provides
fermions in the adjoint of SO(32). NS and R sector together yield an N = 1 SO(32)
vector multiplet. The list of consistent closed string theories in ten dimensions (table
2.1) is supplemented by the ten dimensional theory containing (unoriented) closed
strings and open strings in table 2.8.
2.4.3.2 Orbifold compactiﬁcation So far, we have studied the consistency conditions implied by a torus compactiﬁcation
of the transverse dimension. Since type I theory is a ten dimensional N = 1 supersymmetric theory, torus compactiﬁcations will result in extended supersymmetries in
lower dimensions (e.g. N = 4 in four dimensions). For phenomenological reasons it
is desirable to obtain less supersymmetry. This can be achieved by taking the transverse space to be an orbifold. In the following we will add Oplanes and Dbranes to
the orbifold compactiﬁcation considered in section 2.2.2. We supplement the Z2 action
(2.2.2.1) with an ΩR action, where R acts on the target space in the same way as given
in (2.2.2.1). Hence, our discrete group is generated by R and ΩR. The third nontrivial
element is the product of the two generators: Ω. Thus, the theory contains O5planes
and O9planes. We expect that we need to add D5branes and D9branes in order to
preserve RR charge conservation. Before, studying the open strings induced by those
Dbranes let us discuss the untwisted and twisted closed string sector states. We focus
on the massless part of the spectrum. All the information needed to ﬁnd the untwisted
massless states is given in table 2.5. In addition to the Z2 symmetry we also need to
respect the Ω and ΩR symmetry. This is done by symmetrization in the NSNS sector
and antisymmetrization in the RR sector. Thus the untwisted NSNS sector contains
the metric Gij , the dilaton Φ and ten scalars. In the RR sector one ﬁnds a selfdual
and an antiselfdual two form and twelve scalars. The relevant twisted sector states
are listed in table 2.6. Taking into account that there are 16 ﬁxed points we obtain
48 massless scalars in the twisted NSNS sector, and 16 massless scalars in the twisted 2. Compactifying the transverse dimensions 117 RR sector. Adding the fermions in the Ω and ΩR invariant combinations of NSR
and RNS sector states, one obtains ﬁnally a d = 6, N = 1 supergravity multiplet, one
tensor multiplet and 20 hypermultiplets. (We should emphasize again that the present
review is not self contained as far as the supergravity representations are concerned.
The reader may view the arrangement of the massless states into multiplets as some
additional information which is not really employed in the forthcoming discussions.
In order to obtain a nice overview about supermultiplets in various dimensions we
recommend[399].)
As already mentioned, we need to add D5 and D9branes in order to cancel the
Oplane RR charges. Thus the ChanPaton matrix is built out of the following blocks:
λ(99) corresponding to strings starting and ending on D9branes, λ(55) corresponding
to strings starting and ending on D5branes, λ(59) and λ(95) corresponding to open
strings with ND boundary conditions in the compact dimensions. The action of Ω
and ΩR on the ChanPaton labels is as described in (2.4.3.2). The γΩ and γΩR posses
also a block structure distinguishing between the action on a string end at a D5 or
(9) D9brane, e.g. γΩ represents the Ω action on a ChanPaton label corresponding to
an open string end on a D9brane. Finally, we specify the representation of the Z2
element R (2.2.2.1) as follows
R: ψ, ij → (γR)ii R (ψ ) , i j
(9) −
γR 1 jj . (2.4.3.24) (5) Also, the γR can be split into two blocks: γR and γR . The requirement that performing twice the same Z2 action should leave the state invariant leads to the conditions
that every gammablock containing an Ω in the subscript must be either symmetric
or antisymmetric, whereas the gammablocks without an Ω in the subscript must
square to the identity58. At this point, we need to discuss a subtlety of the ﬁvenine
sector, i.e. strings with ND boundary conditions along the compact dimensions. In
that sector the Fock space state (without the ChanPaton label) has an Ω2 and an
(ΩR)2 eigenvalue of minus one. Unfortunately, we did not develop the techniques
needed to show this, in this review. An argument employing an isomorphism between
the algebra of vertex operators and Fock space states can be found in[204]. Since Ω2
and (ΩR)2 should leave states invariant, this minus sign needs to be canceled by an
(9) appropriate action on the ChanPaton labels. For example a symmetric γΩ implies
(5)
(5)
(9)
an antisymmetric γΩ , and a symmetric γΩR implies an antisymmetric γΩR .
Let us now study the amplitudes in the loop channel. For strings starting and ending on D9branes there is a tower of KaluzaKlein momentum modes but no winding
modes. The D9branes are space ﬁlling and thus must lie on top of each other. Open
58 We ﬁx a possible phase to one. 2. Compactifying the transverse dimensions 118 strings with both ends on a D5brane can have winding modes and can be separated
in the compact directions. (In our previous computation we did not consider this separation, because it is not relevant in the tree channel infrared limit. In order to see
explicitly that the dependence on the distance among D5branes drops out, we will
take it into account, here. We call the component of the position of the ith brane ca .)
i
Further, all amplitudes obtain an additional insertion
1+R
,
2 (2.4.3.25) ensuring that we trace only over states which are invariant under the orbifold group.
For terms containing an R insertion the D5brane must be located at R ﬁxed points,
since otherwise the states in the (55) sector are not eigenstates under R. For the same
reason the winding or KaluzaKlein momentum modes have to vanish in the presence
of an R insertion. Further, there will be additional signs for oscillators pointing into
a compact dimension. This modiﬁes the oscillator contribution to the trace in a
straightforward way. (We leave the details as an exercise.) Taking into account all
these eﬀects and the discussion in section 2.3.2 one ﬁnds for the annulus amplitude A 4
∞
8
2
f4 e−πt dt
V6
−3
(9) 2
− 2πtn
2 8π α
eρ
tr γ1
A=−
8
4
t4
f1 (e−πt ) n=−∞ √
2
9
∞
2πw ρα +ca −ca ) (
i
j
(5)
(5)
2πα
e−t
γ1
+
γ1 jj
ii
a=6 w =−∞ i,j ∈5 −2 4
4
f2 e−πt f4 e−πt
4
4
f1 (e−πt ) f3 (e−πt ) f4
+4 34
f1 −πt 4
e
f4
4
(e−πt ) f2 −πt e
(e−πt ) 16 (5) (9) tr γR,I tr γR I =1
(9) 2
tr γR 16 + (5) tr γR,I 2 , (2.4.3.26) I =1 where we have formally assigned a gamma with subscript 1 to the action of the identity
element of the orbifold group on the ChanPaton labels. The sum over i, j ∈ 5 means
that we sum over all ChanPaton labels belonging to an open string end on a D5brane.
The index I = 1, . . . , 16 labels the ﬁxed 5planes, and a corresponding subscript at a
γ (5) indicates that the D5brane is located on the I th ﬁxed plane.
Next, we want to compute the Klein bottle amplitude K. It contains the insertions
Ω and ΩR. In principle, we have to take the trace over untwisted and twisted sector
states (with the (−)F insertion). Because half of the RR sector states have the opposite
(−)F eigenvalue than the other half, RR sector states do not contribute to the trace
with a (−)F insertion. The same applies to RR and NSNS twisted sector states.
Eigenstates of Ω have zero winding numbers whereas for eigenstates of ΩR the Kaluza 2. Compactifying the transverse dimensions 119 Klein momenta are zero. With this ingredients we ﬁnd
K = −8 dt
8π 2α
t4 ∞
e−2πt V6
4 8
f4
8
f1 (e−2πt ) −3
2
− πtn
ρ e 4 ∞ + n=−∞ 4 −πtρw 2 e w =−∞ . (2.4.3.27) Finally, for the M¨bius strip amplitude M we need to trace over R sector states
o
with an Ω + ΩR insertion. Eigenstates correspond to open strings starting and ending
on the same brane. According to our earlier assignments, the ΩR eigenvalue of the R
vacuum corresponding to a string ending on a D5brane is minus one, and so is the
Ω eigenvalue of the R vacuum corresponding to a string ending on D9branes. To
determine the remaining eigenvalues one has to act with R on the Ramond vacuum.
R can be viewed as a 180◦ rotation and the R vacua as target space spinors. Hence,
half of the Ramond vacua have R eigenvalue minus one and the other half plus one.
For this reason, only D9branes contribute to the term with the Ω insertion whereas
only D5branes give a nonvanishing result for the trace containing an ΩR insertion.
The result for the M¨bius strip is
o
M= 8
−2πt f 8 e−2πt
V6
dt
−3 f2 e
4
8π 2α
8
8
4
t4
f1 (e−2πt ) f3 (e−2πt ) ∞ 2
(9) T
(9) −1
− 2πtn
eρ
γΩ
tr γΩ 4 n=−∞ (5) −1
γΩR +tr ∞ (5) T
γΩR −2πtρw 2 e w =−∞ 4 . (2.4.3.28) With the next steps necessary to compute the total RR charge of the system we
1
1
are familiar by now. We replace t = 2l in the annulus, t = 4l in the Klein bottle
1
and t = 8l in the M¨bius strip. In order to be able to read oﬀ the infrared (large l)
o
asymptotics we use formulæ (2.3.2.22) and (2.4.3.12). The ﬁnal result is
A + K + M −→ − V6
4 ρ2
+ dl 4π 2α
l→∞ 1
ρ2 1
+
4 (9) 2 trγ1 16 (9) (9) −1 − 32tr (5) 2 trγ1 3 γΩ − 32tr
(5) trγR + 4trγR,I
I =1 (γΩ )T (5) −1 γΩR
2 . (γΩ )T + 322
+ 322
(2.4.3.29) 2. Compactifying the transverse dimensions 120 The setup respects RR charge conservation if (2.4.3.29) vanishes. Thus, we need 32
D9branes and 32 D5branes. (A gamma representing the identity is of course the
identity matrix.) Further, we take
(5) T (5) γΩR = γΩR (9) T (9) , γΩ = γΩ . (2.4.3.30) Our previous discussion of the (59) sector implies
(5) T (5) γΩ = − γΩ (9) T (9) , γΩR = − γΩR . (2.4.3.31) The remaining representation matrices can be found by imposing that the gammas
should form a projective59 representation of the orientifold group (Z2 × Z2 ). We
simply choose
(5) = γΩR γΩ (9) = γΩR γΩ . γR γR (5) (5) (2.4.3.32) (9) (9) (2.4.3.33) By ﬁxing a basis in the ChanPaton labels we obtain
(5) (9) γΩR = γΩ = I, (2.4.3.34) where the rank of the identity matrix is 32. The antisymmetric form is
(5) (9) γΩ = γΩR = 0 iI
−iI 0 , (2.4.3.35) with I being a 16 × 16 identity matrix, here. Note that our choice is consistent with
(·)
the requirement that γR squares to the identity. So far, we did not take into account
(·)
that the last term in (2.4.3.29) has to vanish. With γR being traceless this is ensured.
We have now all the ingredients needed to determine the open string spectrum. Let
us ﬁrst study strings starting and ending on the D9branes, or in short the (99) sector.
We keep states which are invariant under each element of the orientifold group. (The
D9branes are ﬁxed under each element of the orientifold group.) In the NS sector
we ﬁnd massless vectors with the ChanPaton matrix
(99) λvector = AS
−S A , (2.4.3.36) where A denotes a real antisymmetric and otherwise arbitrary 16 × 16 matrix and S
stands for a real 16 × 16 symmetric matrix. For the scalars in the NS sector one ﬁnds
(99) λscalars = A1 A2
A2 −A1 , (2.4.3.37) 59
“Projective” means up to phase factors, which drop out since the gamma acts in combination
with its inverse on the ChanPaton label. 2. Compactifying the transverse dimensions 121 where the Ai are 16 × 16 antisymmetric matrices. Let us ignore the D5branes for
a moment and determine the gauge group and its action on the scalars in the (99)
sector. Since the vectors are in the adjoint of the gauge group, the gauge group should
be 162 dimensional. U (16) is a good candidate. Further, we know that the vector
should transform in the adjoint under global gauge transformations under which it
should not change the form speciﬁed by (2.4.3.36). Thus, we deﬁne an element of the
gauge group as
g (9) = exp Ag Sg −Sg Ag , (2.4.3.38) where Sg (Ag ) are real anti(symmetric) matrices with inﬁnitesimal entries. A gauge
transformation acts on the ChanPaton matrix as
λ(99) → g (9)λ(99) g (9) −1 = Ag Sg
−Sg Ag ,λ . (2.4.3.39) We observe that the vectors transform in the adjoint and the form of the ChanPaton
matrix is preserved. Note also that g (9) is unitary and has 162 parameters. It is a
U (16) element. The U (1) subgroup corresponds to Ag = 0 and Sg proportional to the
identity. From our assignment that the ChanPaton matrix transforms in the adjoint
of U (16) it is also clear that the ChanPaton label i and j transform in the 16 and
16 of U (16) Thus, the scalars can be decomposed into the antisymmetric 120 + 120.
One may also explicitly check that the form of the ChanPaton matrix for the scalars
is not altered by a gauge transformation. We leave the discussion of the fermions in
the R sector as an exercise. The result is that that half of them carry the ChanPaton
matrix (2.4.3.36) and the other half the matrix (2.4.3.39). Altogether the (99) sector
provides a vector multiplet in the adjoint of U (16) and a hypermultiplet in the 120
+ 120.
Now, we include the D5branes. Here, we have to distinguish between the case
that a D5brane is situated at a ﬁxed plane or not. In the ﬁrst case the (55) strings
have to respect the ΩR and R symmetry, whereas in the second case these orientifold
group elements just ﬁx the ﬁelds on the image brane. Suppose we have 2mI D5branes
at the I th ﬁxed plane. (The number of the D5branes per ﬁxed plane must be even,
since otherwise they cannot form a representation of the orientifold group.) The NS
sector leads again to a massless vector and massless scalars with almost the same ChanPaton matrices as in the (99) sector ((2.4.3.36) and (2.4.3.39)). The only diﬀerence is
that the antisymmetric and symmetric matrices are now mI × mI instead of 16 × 16. Hence, we obtain a vector multiplet in the adjoint of U (mI ) and a hypermultiplet in
the mI (mI 1) + mI (mI 1).
2
2 2. Compactifying the transverse dimensions 122 Let 2nj D5branes be situated away from the ﬁxed plane (but on top of each other).
For the (55) sector belonging to those D5branes we impose invariance under Ω, only.
(5) The solution for γΩ is given in (2.4.3.35). Together with the minus eigenvalue on the
massless NS sector Fock space state, this leads to the result that the vector in the (55)
sector is an element of the U Sp (2nj ) Lie algebra in the adjoint representation. Taking
into account (part of) the R sector this is promoted to a U Sp (2nj ) vector multiplet.
The scalars together with the remaining R sector states form a hypermultiplet in the
antisymmetric nj (2nj 1) representation.
It remains to study the (95) sector. (Here, one has to take into account that
along the compact directions NS sector fermions are integer modded whereas R sector
fermions are half integer modded. This is quite similar to the twisted sector closed
string. In particular, the (95) NS sector ground state is already massless. Hence,
the NS sector does not give rise to massless vectors. We do not impose Ω or ΩR
invariance on (95) strings since they are mapped onto (59) strings by the worldsheet
parity inversion. If the considered D5branes are situated at one of the ﬁxed planes
we impose R invariance. In this case, one ﬁnds in the NS sector two scalars with the
ChanPaton matrix
λ(95) = X1 X2
−X2 X1 , (2.4.3.40) where the Xi are general mi × 16 matrices. Together with the R sector this leads to
a hypermultiplet in the (16 , mI ) of U (16) × U (mI ), (the hypermultiplet is neutral
under the gauge group living on D5branes not situated at the I th ﬁxed plane). For
D5branes which are not a ﬁxed plane the (95) sector provides a hypermultiplet in
the (16, 2nJ ) of U (16) × U Sp (2nj ).
Altogether we ﬁnd the gauge group is
16 U (16) × I =1 U (mI ) × U Sp (2nj ) , (2.4.3.41) J where j labels the D5brane packs away from ﬁxed planes. In addition the total
number of D5branes has to be 32 (images are counted), i.e.
16 2mI + 2
I =1 2nj = 32. (2.4.3.42) j There are hypermultiplets in the representation
16 2 (120 , 1, 1) +
I =1 + 1
2 1, mI (mI 1) , 1
2 I + (16, mI , 1)I (1, 1, nj (2nj 1) 1)j + (1, 1, 1) + (16, 1, 2nj )j ,
j (2.4.3.43) 2. Compactifying the transverse dimensions 123 where we have split the antisymmetric representation of U Sp (2nj ) into its irreducible
parts and an index I , j refers to the gauge group on the D5brane pack at a ﬁxed plane
and oﬀ a ﬁxed plane, respectively. It can be checked that the eﬀective six dimensional
theory is free of anomalies. The models belonging to diﬀerent distributions of the
D5branes on and oﬀ ﬁxed planes can be continuously transformed into each other.
In the ﬁeld theory description this corresponds to the Higgs mechanism.
We have seen that adding to the orbifold compactiﬁcation of section 2.2 Oplanes
and Dbranes gives a very interesting picture. Apart from the closed string twisted
sector states located at the orbifold ﬁxed planes we obtain various ﬁelds from open
strings ending on Dbranes. These Dbranes can be moved within the compact directions while keeping the geometry ﬁxed. The techniques described in this section can be
also applied to phenomenologically more interesting setups leading to four dimensional
theories. A description of such models is beyond the scope of the present review. Chapter 3 NonPerturbative description of
branes
3.1 Preliminaries In the previous sections we gave a perturbative description of various extended objects: the fundamental string, orbifold planes, Dbranes and Orientifold planes. The
string plays an outstanding role in the sense that ﬁeld theories on the worldvolumes
of the other extended objects are eﬀective string theories. The quantization of the
fundamental string is performed in a trivial target space (i.e. the target space metric
is the Minkowski metric and all other string excitations are constant or zero). Further,
the worldsheet topology is speciﬁed to the spherical (for closed strings) or disc (for
open strings) topology (after Wick rotating to Euclidean worldsheet signature). Our
treatment leads to a perturbative expansion in the genus of the worldsheet (see section
2.1.4). The perturbative expansion is governed by the string coupling
gs = e Φ , (3.1.0.1) which needs to be small. Perturbative closed string theory has an eﬀective ﬁeld theory
description which contains supergravity. How does one obtain insight into regions
where gs is large? Clearly, the perturbation theory breaks down in this case, and
indeed this region is rather diﬃcult to study. There are, however, a few results one
can obtain also for strong couplings. Let us recall how non perturbative eﬀects in YangMills theory can be studied. Apart from the trivial vacuum, (Euclidean) YangMills
theory contains several other stable vacua, viz. the instantons. Studying ﬂuctuations
around an instanton vacuum, one ﬁnds an additional weight factor in the path integral
which comes from the background value of the action and is of the form
− e n
g2 124 , 3. Universal Branes 125 where n is the instanton number and g is the YangMills gauge coupling. As long as g is
small, the ﬂuctuations around an instanton vacuum are heavily suppressed. However,
as soon as g becomes large, the suppression factor becomes large. Thus, knowing about
the instanton solutions in YangMills theory gives a handle on non perturbative eﬀects.
But how can one know, that one does not have to include strong coupling eﬀects into
the theory before deriving the instanton solutions? The answer is that instantons are
stable, they are characterized by a topological number which cannot be changed in
a continuous way when taking g from small to large. Therefore, instantons can give
information about strongly coupled YangMills theory even though they are found as
solutions to the perturbative formulation of YangMills theories. States (vacua) with
such a feature are called BPS states.12
Therefore, our aim will be to ﬁnd BPS states in string theory. In the low energy
limit, the various superstring theories are described by supergravities. Insights into
nonperturbative eﬀects in string theory can be gained by ﬁnding the BPS states of
perturbative string theory. As a guiding principle, we will look for solutions to the
eﬀective equations of motion that preserve part of the supersymmetry (i.e. are invariant
under a subset of the supersymmetry transformations). Roughly speaking, it is then
the number of preserved supersymmetries which cannot be changed continuously when
taking the string coupling from weak to strong. We will see that such solutions can
be viewed as branes. The number of branes takes the role of the instanton number
in the YangMills example discussed above. We will be very brief in our analysis and
essentially only summarize some of the important results. The classical review on
branes as supergravity solutions is[152] and we will give more references in the end of
this review. 3.2 Universal Branes From section 2.1.4 we recall that all the closed superstring theory eﬀective actions
contain a piece
Suniv = 1
2κ2 √
1
d10x −Ge−2Φ R + 4 (∂ Φ)2 − H 2 .
12 (3.2.0.1) In the present section we will truncate all closed string eﬀective ﬁeld theories to (the
supersymmetric extension of) (3.2.0.1). This is consistent because we will restrict on
backgrounds where the discarded part of the action vanishes and the corresponding
equations of motion are satisﬁed trivially. By adding appropriate terms including
1
There are also stable non BPS states (for reviews see e.g. [424, 319, 195]). We will not discuss
these.
2
BPS stands for the names Bogomolny, Prasad and Sommerﬁeld, and refers to the papers[76, 379]. 3. Universal Branes 126 fermions (2.1.4.5) can be promoted to an N = 1 supersymmetric theory. (For type II
theories this is a subsymmetry of the N = 2 supersymmetry.) The supersymmetric
extension is usually given in the Einstein frame. The action (3.2.0.1) is written in the
string frame were the string tension is a constant and independent of the dilaton. The
Einstein frame is obtained by the metric redeﬁnition
Φ gµν = e− 2 Gµν , (3.2.0.2) where Gµν is the string frame metric and gµν is the Einstein frame metric. The action
(3.2.0.1) takes the form
SE,univ = 1
2κ2 √
1
1
d10x −g R − (∂ Φ)2 − e−Φ H 2 .
2
12 (3.2.0.3) We observe that (3.2.0.3) starts with the familiar Einstein Hilbert term (therefore the
name “Einstein frame”). Further, the kinetic term of the dilaton has the “correct”
sign now, and the coupling of the B ﬁeld is Φ dependent. In the supersymmetric
extension, a gravitino and a dilatino are added. We do not give the supersymmetric
action explicitly. For us, it will suﬃce to know the supersymmetry transformations of
the gravitino and the dilatino. These are
1 −Φ
ν
e 2 Γµ νρκ − 9δµ Γρκ Hνρκ ,
96
Φ
1
1
δλ = − √ Γµ ∂µ Φ + √ e− 2 Γµνρ Hµνρ ,
22
24 2 δψµ = Dµ + (3.2.0.4)
(3.2.0.5) where ψµ denotes the gravitino and λ the dilatino. The Gamma matrices with curved
indices are obtained from ordinary Gamma matrices (16 × 16 matrices satisfying the
usual Cliﬀord algebra in ten dimensional Minkowski space) by transforming the ﬂat
index with a vielbein to a curved one. A Gamma with multiple indices denotes the
antisymmetrized product of Gamma matrices. The spinor is the supersymmetry
transformation parameter.
Sometimes it is useful to formulate the theory in a slightly diﬀerent way. To
this end, one adds to the action (3.2.0.3) a Lagrange multiplier term providing the
constraint of a fulﬁlled Bianchi identity. Calling the Lagrange multipliers Aµ1 ...µ6 ,
such a term looks like
d10x µ1 ...µ10 Aµ1 ...µ6 ∂µ7 Hµ8 µ9 µ10 . (3.2.0.6) The Aµ1 ...µ6 equation of motion yields the Bianchi identity of the B ﬁeld strength
Hµνρ . However, one can alternatively solve the B ﬁeld equation of motion with the
result
Hµνρ ∼ eΦ µνρµ1 ...µ7 K µ1 ...µ7 , (3.2.0.7) 3. The fundamental string 127 with
Kµ1 ...µ7 = ∂[µ1 A µ2 ...µ7 ]. (3.2.0.8) This means that we can trade the antisymmetric tensor B for a six form potential
A. Choosing an appropriate normalization for the Lagrange multiplier terms (3.2.0.6),
the eﬀective action (3.2.0.3) in terms of the six form potential A reads
1
˜
SE,univ = 2
2κ √
1 Φ2
1
eK .
d10x −g R − (∂ Φ)2 −
2
2 · 7! (3.2.0.9) Also in this form the action can be supersymmetrized. In terms of the six form
potential A the gravitino and dilatino supersymmetry transformations read
1Φ
ν
e 2 3Γµ ν1 ...ν7 − 7δµ1 Γν2 ...ν7 Kν1 ...ν7 ,
2 · 8!
Φ
1
1
√
δλ = − √ Γµ ∂µ Φ −
e 2 Γµ1 ...µ7 Kµ1 ...µ7 .
22
2 · 2 2 · 7! δψµ = Dµ + (3.2.0.10)
(3.2.0.11) In the following two subsections, we will present two solutions preserving half of
the supersymmetry. 3.2.1 The fundamental string The solutions we are going to discuss in the present and subsequent sections are generalizations of extreme Reissner–Nordstr¨m black holes. Reissner–Nordstr¨m black
o
o
holes are solutions of Einstein–Hilbert gravity coupled to an electro magnetic ﬁeld.
They carry mass and electric or magnetic charge. Extreme Reissner–Nordstr¨m black
o
holes satisfy a certain relation between the charge and the mass. (In our case such
a relation will be dictated by the requirement of partially preserved supersymmetry.)
Replacing the electro magnetic ﬁeld strength F by its dual F interchanges electric
with magnetic charge. (For a more detailed discussion of Reissner–Nordstr¨m black
o
holes see e.g.[442].)
The action (3.2.0.3) bears some analogy to four dimensional Einstein gravity coupled to an electro magnetic ﬁeld. The diﬀerence is that the theory is ten dimensional
instead of four dimensional and the electro magnetic ﬁeld strength is replaced by the
three form H . In addition, there is the scalar Φ. Since the gauge potential is now a
two form which naturally couples to the worldvolume of a string, we look for “extreme
Reissner–Nordstr¨m black strings” instead of black holes. The corresponding ansatz
o
for the metric is
ds2 = e2A ηij dxi dxj + e2B δab dy ady b , (3.2.1.1) 3. The fundamental string 128 with i, j = 0, 1 and a, b = 2, . . . , 9. Further, A and B are functions of
r= δab y a y b , (3.2.1.2) only. Here, we have taken the second step before the ﬁrst one in the sense that
ﬁrst we should have thought about what kind of isometries we would like to obtain
and only afterwards we should have written down a general ansatz respecting the
isometries. Therefore, let us perform the ﬁrst step now and discuss the isometries of
the ansatz. Clearly, there is an SO (1, 1) isometry acting on the xi . This means, that
up to Lorentz boosts, xi spans the worldvolume of a straight static string. There is no
further dependence on xi in the ansatz because we do not wish to distinguish a point
on the worldvolume of the string. The other isometry acts as an SO (8) on the y a .
This is the natural extension of the SO (3) isometry associated with nonrotating four
dimensional black holes. It is SO (8) now because the space transverse to the string is
eight dimensional (whereas in 4d black holes the space transverse to the hole is three
dimensional). The r dependence respects the SO (8) isometry. Distinguishing between
diﬀerent values of r means specifying the position of the string, i.e. r measures the
radial distance from the string.
In order not to spoil the above symmetries, we choose for the remaining ﬁelds the
ansatz
B01 = −eC , Φ = Φ (r) , (3.2.1.3) where C is also a function of r only. All other components of B are zero. Viewed as a
two form, B is proportional to the invariant volume form of the string worldvolume.
The factor eC may depend on r.
The ansatz for the target space spinors is that they all vanish. As mentioned earlier
we are interested in situations where the solution preserves part of the supersymmetry
because this ensures that we can continuously take the string coupling from weak to
strong. In particular, the unbroken supersymmetry is parameterized by spinors for
which the gravitino and dilatino values of zero do not change under supersymmetry
transformations, i.e. those for which the rhs of (3.2.0.4) and (3.2.0.5) vanish. In order
to ﬁnd such solutions for our ansatz it is convenient to represent the ten dimensional
Gamma matrices A, B = 0, . . . , 9,
{ΓA , ΓB } = 2ηAB (3.2.1.4) as a tensor product of 2 × 2 matrices γi in 1 + 1 and 8 × 8 matrices Σa in 8 dimensions3
ΓA = (γi ⊗ I, γ3 ⊗ Σa ) ,
3 The corresponding algebras are {γi , γj } = 2ηij and {Σa , Σb} = 2δab . (3.2.1.5) 3. The fundamental string 129 where I is the 8 × 8 identity matrix and
γ3 = γ0γ1 (3.2.1.6) squares to the 2 × 2 identity matrix. Further, we have to take into account that the
ten dimensional N = 1 supersymmetry parameter is subject to the constraint
Γ11 = . (3.2.1.7) Under certain conditions to be speciﬁed below the variations of the gravitino and the
dilatino vanish for
=e 3Φ
8 ε0 ⊗ η 0 , (3.2.1.8) where ε0 and η0 are SO (1, 1) and SO (8) constant spinors, respectively, which satisfy
the lower dimensional chirality conditions
9 (1 − γ3) ε0 = 0 , 1− Σa η0 = 0. (3.2.1.9) a=2 This breaks the supersymmetry to half the amount of the perturbative (trivial) vacuum. (The condition (3.2.1.7) could be also satisﬁed by choosing simultaneously the
opposite chiralities in the two equations (3.2.1.9).)
We already mentioned that only under certain conditions we can ﬁnd unbroken
supersymmetries at all. Requiring that asymptotically (r → ∞) we obtain the perturbative vacuum, these conditions read
3
(Φ − Φ0 ) ,
4
1
B = − (Φ − Φ0) ,
4
3
C = 2Φ − Φ0
2
A= (3.2.1.10)
(3.2.1.11)
(3.2.1.12) where Φ0 is the asymptotic value of Φ. Hence supersymmetry leaves only one function
out of our ansatz undetermined. This function can be taken to be the dilaton whose
equation of motion boils down to
δ ab∂a ∂b e−2Φ(r) = 0, (3.2.1.13) i.e. the “ﬂat” Laplacian of the transverse space (spanned by the y a) acting on e−2Φ
has to vanish. As in the case of four dimensional black holes, we solve this equation
everywhere but at the origin r = 0, where there are additional contributions due to
a source string. (We do not add the source string explicitly here, but will infer its 3. The fundamental string 130 properties (tension and charge) in an indirect way below. For the explicit inclusion of
the source term see e.g.[152].) The solution to (3.2.1.13) reads
e−2Φ = e−2Φ0 1+ k
r6 , (3.2.1.14) where k is an integration constant which will be related to the string tension below.
Plugging this back into (3.2.1.10) – (3.2.1.12) and in the ansatz gives the ﬁnal solution.
Next, we would like to deduce the tension of the string source from our solution.
This is done by studying the Newtonian limit of general relativity. In particular, by
comparing the Einstein equation with the geodesic equation of a point particle (which
has constant mass in the Einstein frame) one ﬁnds that the Newton potential of the
stringsource is encoded in the subleading term in a large r expansion of g00. Therefore,
we ﬁrst observe that for large r
g00 = −1 + 3k
+ ... .
4r6 (3.2.1.15) The relation between g00 and the Newton potential of a string is explicitly such that4
3
δ (r)
1
∂ r7 ∂r g00 = − κ2 TE
7r
r
2
Ω7r7 (3.2.1.16) holds, with the understanding that terms denoted by . . . in (3.2.1.15) are neglected.
The string tension is denoted by TE . Further, the unit volume of a seven–sphere Ω7
enters the expression. Hence, we obtain
TE = 3k
Ω7 .
κ2 (3.2.1.17) We put the index E at the tension in order to indicate that it is measured in the Einstein frame. What we are actually interested in, is the tension in the string frame. This
is readily obtained by noticing that transforming back to the string frame (asymptotically) implies
κ2 → e2Φ0 κ2 , Ω7 → e 7Φ0
2 Ω7 . (3.2.1.18) Thus in the string frame the tension is5
T=
4 3k 3Φ0
e 2 Ω7 .
κ2 (3.2.1.19) Equation (3.2.1.16) has the same form as the equation satisﬁed by the Newton potential. The
3
numerical factor of 2 = 7−p on the rhs of (3.2.1.16) is a matter of convention, which ﬁxes the relation
4
between κ, the speed of light (c = 1), and Newton’s constant (see e.g.[432]). We choose our convention
in agreement with[152], where explicit source terms (containing the tension) are added.
5
On dimensional grounds, one would expect a diﬀerent scaling of T . In order to obtain this, one
has to take into account that k is a dimensionful quantity. In (3.2.1.19) k is given Einstein frame
units. 3. The NS ﬁve brane 131 1
Recalling that our “elementary particle” is a string of tension 2πα and requiring that
any string like object must consist out of an integer number N of elementary strings we ﬁnally determine the integration constant k to be6
k=N 3Φ0
κ2
e− 2 .
6πα 7 (3.2.1.20) It remains to compute the U (1) charge carried by the vacuum. This is basically
done by integrating the B equation of motion over the transverse space. The result is
1
µ= √
2κ e−Φ H, (3.2.1.21) S7 where the integration is over an asymptotic sevensphere enclosing the string source.
The U (1) charge is denoted by µ. Expressing the result in terms of the string tension
one obtains
µ= √
N
2κ
.
2πα (3.2.1.22) This equality is related to partially unbroken supersymmetry. If the conﬁguration
was not stable the tension of the bound state would be larger than the sum of the
elementary tensions. Hence, the rhs of (3.2.1.22) is larger for general (non BPS) states.
The BPS state saturates a general inequality. Since the BPS state is stable, there can
be no state with less tension and the same charge since otherwise the BPS state would
decay into such a state. The lower bound on the tension set by the BPS state is called
the Bogomolnyi bound. 3.2.2 The NS ﬁve brane In this subsection we repeat the analysis of the previous section, however, with the
action (3.2.0.9) instead of (3.2.0.3). Thus, we will obtain the magnetic dual of the
previously discussed string solution. This is called the NS ﬁve brane. Its properties
(tension, charge) will be ﬁxed in terms of the string properties via the Dirac quantization condition. (For generalizations of the Dirac quantization condition to extended
objects see[353, 438].) As the derivation of the NS ﬁve brane solution goes along the
same lines as the one given in the previous subsection, we will be even more sketchy
here. Instead of the two form potential, we have now the six form potential A. Since
an object which extends along ﬁve spatial dimensions naturally couples to a six form
potential, we choose the following ansatz for the metric
ds2 = e2A ηij dxi dxj + e2B δab dy ady b , (3.2.2.1) 6
Here we use the fact that for a superposition of BPS states there is no binding energy, i.e. the
total tension is obtained by simply summing the tensions of the individual BPS states. 3. The NS ﬁve brane 132 where now i, j = 0, . . . , 5 and a, b = 6, . . . , 9. The ﬁve brane worldvolume extends
along the xi directions and the functions A and B are allowed to depend on the radial
distance from the ﬁve brane r with
δab y a y b . (3.2.2.2) A012345 = −eC , (3.2.2.3) r=
The ansatz for the six form potential is where C is a function of r. The components of A which cannot be obtained by
permuting the indices in (3.2.2.3) are zero. The ﬁnal input is that also the dilaton
depends only on r,
Φ = Φ (r) . (3.2.2.4) All fermionic ﬁelds are again set to zero. There is an unbroken supersymmetry if
we can ﬁnd a spinor such that the gravitino and dilatino transformations (3.2.0.10)
and (3.2.0.11) vanish. It turns out that half of the supersymmetry is preserved if the
following relations hold
1
A = − (Φ − Φ0) ,
4
3
B=
(Φ − Φ0 ) ,
4
3
C = −2Φ + Φ0 ,
2 (3.2.2.5)
(3.2.2.6)
(3.2.2.7) where Φ0 denotes again the asymptotic r → ∞ value of the dilaton. (In addition
to partially unbroken supersymmetry we have once again imposed that for large r
the vacuum should approach the perturbative vacuum.) Under these conditions the
equations of motion boil down to
δ ab ∂a ∂b e2Φ = 0. (3.2.2.8) We solve this equation everywhere but at r = 0 where we allow for additional contributions due to source terms. One ﬁnds
e2(Φ−Φ0 ) = 1 + ˜
k
.
r2 (3.2.2.9) ˜
The integration constant k can be ﬁxed by exploiting the Dirac quantization condition.
To this end we compute the charge carried by the vacuum
√
˜
1
2Ω3k Φ0
Φ
e
K=
µ= √
˜
e2,
(3.2.2.10)
κ
2κ S 3 3. The NS ﬁve brane 133 where the integral is over an asymptotic three–sphere surrounding the ﬁve brane and
Ω3 denotes the volume of a unit three–sphere. Now, the Dirac quantization condition
reads
˜
µµ = 2π NN,
˜ (3.2.2.11) where µ is the charge of N elementary strings (3.2.1.22). The number of ﬁve branes
˜
(number of magnetic charges) is N . This ﬁxes the integration constant,
˜
π N − Φ0
˜
e 2,
k=
TS Ω3 (3.2.2.12) 1
where TS is the elementary string tension (TS = 2πα in the string frame). By computing the gravitational potential in the Newtonian limit, one ﬁnds the tension of the ﬁve brane in the Einstein frame (with TS and κ also in Einstein frame units)
˜
π N − Φ0
˜
e 2.
TE =
TS κ2 (3.2.2.13) ˜
The mass dimension of TE is six. Hence, we obtain
˜
2πα π N
˜
T = e−2Φ0
κ2 (3.2.2.14) 2
in the string frame. We observe that the ﬁve brane tension behaves as 1/gs . In the perturbative region the NS ﬁve brane is very heavy whereas it becomes lighter when
the string coupling increases.
The NS ﬁve brane is an extended object for which we did not give a perturbative
description. Indeed, such a description is not known. One could try to quantize strings
in the NS ﬁve brane background. This is possible only in certain spatial regions.
Firstly, for large r the background becomes ﬂat, and we know how to quantize strings
there. But also in the background at r → 0 (the near horizon limit) one can ﬁnd a
quantized string theory. In that limit the string frame metric reads
˜
˜
ds2 = ηij dxi dxj + k (d log r)2 + kdΩ2,
s
3 (3.2.2.15) and the dilaton is linear in ρ = log r. With dΩ2 we denote the metric of a unit three–
3
sphere. The NSNS ﬁeld strength H is a constant times the volume element dΩ3. The
geometry factorises into a 5 + 1 dimensional Minkowski space times the direction on
which the dilaton depends linearly times an S 3. Since S 3 is an SU (2) group manifold,
string theory can be quantized in such a background. For more details see[97] or the
review[96]. 3. Type II branes 3.3 134 Type II branes Like in the previous sections, we are interested in setups where only a truncated version
of the eﬀective actions (see section 2.1.4 ) is relevant. The bosonic part of the truncated
type II action reads
S= 1
2κ2 d10x e−2Φ R + 4 (∂ Φ)2 − 1
,
F2
2 (p + 2)! p+2 (3.3.0.1) where Fp+2 denotes the ﬁeld strength of an RR p + 1 form potential. For type IIA p is
even whereas it is odd for type IIB theory. For p = 3 the action has to be supplemented
by the constraint that the ﬁeld strength is selfdual. The (p +2 form) ﬁeld strengths are
not all independent but related by Hodge duality to the ﬁeld strength corresponding
to 6 − p (an 8 − p form ﬁeld strength). We will restrict the discussion to the cases 0 ≤ p < 7. For p = 7 the solution presented in[225] is relevant. The 8brane appears
as a solution of massive type IIA supergravity[56]. How this is related to string theory
(or rather Mtheory) is discussed in the recent paper[236] (see also references therein).
We will consider only a single relevant p at a time. The ﬁeld redeﬁnition (3.2.0.2)
takes us to the action in the Einstein frame
√
3−p
1
1
1
2
e 2 Φ Fp+2 .
(3.3.0.2)
SE = 2 d10x −g R − (∂ Φ)2 −
2κ
2
2 (p + 2)!
The pbrane ansatz reads
ds2 = e2A ηij dxi dxj + e2B δab dy ady b , (3.3.0.3) with i, j = 0, . . . , p and a, b = p + 1, . . . , 9. A and B are functions of
r= δab y a y b (3.3.0.4) The dilaton is also taken to be a function of r. Let us ﬁrst exclude the case p = 3 from
the discussion. For the p + 1 form gauge ﬁeld we choose
A0,... ,p = −eC , (3.3.0.5) where C is a function of r, and all other components of A (which cannot be obtained
by permuting the indices in (3.3.0.5)) are zero. All the other ﬁelds (NSNS B ﬁeld, the
remaining RR forms, and the fermions) are zero. The BPS condition leaves one out
of the four functions A, B , C and Φ undetermined. Choosing for convenience C to be
the undetermined function, these conditions read
7−p
(C − C0 ) ,
16
p+1
B=−
(C − C0 ) ,
16
p−3
4C0
Φ=
(C − C0 ) +
,
4
p−3
A= (3.3.0.6)
(3.3.0.7)
(3.3.0.8) 3. Type II branes 135 where again the boundary condition that for r → ∞ the background should be trivial
has been imposed. C0 denotes the asymptotic value of C which is related to the
asymptotic dilaton value
C0 = p−3
Φ0.
4 (3.3.0.9) The equations of motion reduce to
δ ab ∂a ∂b e−C = 0. (3.3.0.10) We solve this by
e−C = e−C0 + kp
r7−p . (3.3.0.11) The RR charge of the vacuum is
1
µp = √
2κ e 3−p
Φ
2 S 8−p 7−p
Fp+2 = √ Ω8−p kp,
2κ (3.3.0.12) where the integration is over an asymptotic (8 − p)–sphere surrounding the p brane,
and Ω8−p is the volume of the unit sphere. Now we try whether we can identify the type II pbranes with the Dbranes discussed in section 2.3. This trial is motivated by
the observation that the Dbranes considered in section 2.3 are also extended objects
carrying RR charge. In section 2.3 we computed the charge of a single Dbrane to be
(see (2.3.2.31) and (2.3.2.29))
µsingle brane =
p √
2π 4π 2α 3−p
2 . (3.3.0.13) Assuming that the vacuum considered in the present section is composed out of an
integer number Np of single Dbranes, we identify (TS denotes the frame dependent
tension of a single fundamental string)
3−p √
π
2κ 4π 2/TS 2
,
kp = N p
(7 − p) Ω8−p (3.3.0.14) µp = Npµsingle brane .
p (3.3.0.15) such that A ﬁrst consistency check is to observe that the Dirac quantization condition
µp µ6−p = 2πNpN6−p (3.3.0.16) is satisﬁed. After we have ﬁxed the integration constant kp the tension of the brane
solution is determined. Because the vacuum considered here and the Dbranes considered in section 2.3 are BPS objects the tension is related to the charge and we expect 3. Type II branes 136 that our tension should be in agreement with (2.3.2.29). Let us nevertheless compute
it explicitly. To this end, we write down the asymptotic expansion of g00
g00 = −1 + 7 − p kp p−3 Φ0
+ ... .
e4
8 r7−p (3.3.0.17) The tension is given via (see also (3.2.1.16))
1 ∂r
r8−p r8−p ∂r g00 = − δ (r)
7−p 2
κ Tp,E
,
4
Ω8−p r8−p (3.3.0.18) where Tp,E denotes the tension in Einstein frame units and it is understood that terms
denoted by dots in (3.3.0.17) are neglected. This yields
Tp,E = p−3
7−p
Ω8−pkp e 4 Φ0 .
2
2κ Since Tp,E has mass dimension p + 1, it receives a factor of e− (3.3.0.19)
p+1
Φ
4 under the trans formation to the string frame. Taking this and (3.3.0.14) into account, we ﬁnd for the
string frame tension
√
Tp = Npe−Φ0 κ π 4π 2α 3−p
2 1
,
κ (3.3.0.20) in agreement with (2.3.2.29). Thus, we found that the pbrane vacuum can be viewed as
consisting out of an integer number of “elementary” (or magnetic) Dbranes considered
in section 2.3.
So far, we have derived this result only in the case p = 3. In the case p = 3, the
condition (3.3.0.8) is changed into
Φ = Φ0 , C0 = 0. (3.3.0.21) The selfduality condition can be imposed by replacing F5 from our ansatz with F5 + F5 ,
F5 → F5 + F5 . (3.3.0.22) k3
.
r4 (3.3.0.23) The solution for C is
e−C = 1 +
The “electric” charge is
1
µ3 = √
2κ S5 4
dA0123 = √ Ω5 k3.
2κ (3.3.0.24) The replacement (3.3.0.22) implies that the solution carries also a magnetic charge
µp = µp . Thus, the Dirac quantization condition yields (N3 is the number of D3˜
branes)
√κ
,
(3.3.0.25)
k3 = N3 π
2Ω5 3. Type II branes 137 such that
µ3 = √ 2πN3. For the tension, one obtains in the string frame
√
−Φ0 π
.
T3 = N3e
κ (3.3.0.26) (3.3.0.27) Thus, also for p = 3 we can consistently assume that the vacuum solution is made out
of an integer number of the D3branes introduced in section 2.3. We will return to the
solution for p = 3 in section 4.3.
We stop our discussion on the appearance of branes as vacua of the eﬀective actions
at this point. We should, however, mention that there are many more conﬁgurations
which can be constructed. For example, one can ﬁnd vacua where branes lie within
the worldvolume of other higher dimensional branes. Such studies conﬁrm the result
of section 2.3.1.3 that supersymmetry is completely broken unless the number of ND
directions is an integer multiple of four.
A ﬁnal remark about the BPS vacua of type I theory is in order. Although we
called this section “type II branes”, the discussion applies to type I theory as well. In
the closed string sector of type I theory the NSNS B ﬁeld is projected out, and thus
there is neither a fundamental string nor an NS ﬁve brane vacuum in the eﬀective
type I theory. On the other hand, the RR two form potential survives the projection.
Hence, type I theory possesses the D1 and the D5 brane vacua. Chapter 4 Applications
In this chapter, we are going to present some applications of the branes discussed so
far. In the following, we will show that branes are a useful tool in supporting duality
conjectures involving an interchange between strong and weak couplings. As a ﬁrst
example we consider dualities among diﬀerent string theories. Thereafter, ﬁeld theory
dualities will be translated into manipulations within certain brane setups. Next, we
want to present the AdS/CFT correspondence – a duality between closed and open
string theory, or in ﬁrst approximation between gravity and gauge theory. Finally, we
argue that branes allow constructions in which the string scale is about a TeV. Such
setups have the prospect of being discovered in the near future. There are many more
applications of branes in theoretical physics. Some of them we will list in chapter 6
containing suggestions for further reading. 4.1 String dualities There are many excellent reviews on string dualities and we do not plan to provide an
introduction into this subject here. We just want to summarize how branes are mapped
among each other under duality transformations. We start by drawing the Mtheory
star in ﬁgure 4.1. The idea behind this picture is that the theories written at the tips
of the “star” are diﬀerent descriptions of one underlying theory called Mtheory. This
underlying theory is not known. It is assumed to possess a moduli space which looks
like ﬁgure 4.1. The picture is supported by evidences for the conjecture that all the
other theories appearing in ﬁgure 4.1 are related by motions in their moduli spaces.
Let us brieﬂy summarize how these theories are connected. We start at the top of
the star (11D SUGRA) and work our way down to the bottom (type I), ﬁrst counter
clockwise. Compactifying eleven dimensional supergravity on an interval (S 1/Z2 )
yields the eﬀective ﬁeld theory of the heterotic E8 × E8 string. The dilaton is re138 4. String dualities 139
11 D SUGRA Heterotic
E8 × E8 Type IIA M
Heterotic
SO (32) Type IIB Type I Figure 4.1: Mtheory star
lated to the length of the interval such that the string coupling is small when the
interval is short. The E8 × E8 ﬁelds live as twisted sector ﬁelds at the ends of the
interval (the orbifold nine planes). If we take the string coupling of the E8 × E8 het erotic string to be strong, 11D supergravity on an interval provides the more suitable
description. The connection between the E8 × E8 and the SO (32) string was already
discussed in section 2.1.5.4. It does not relate strong with weak coupling but small
with large compactiﬁcation radii in nine dimensions. The heterotic SO (32) string is
connected to type I strings by a strong/weak coupling duality. Now, let us go back
to the top of the star and go down clockwise. Type IIA supergravity can be obtained
by compactifying 11 dimensional supergravity on a circle. The radius of the circle
determines the vev of the dilaton. For small string coupling the circle is small, and
for strong coupling it is large. The connection between type IIA and type IIB strings
is seen by compactifying further down to nine dimensions and inverting the radius, as
argued in section 2.1.5.4. Type I theory is obtained by gauging worldsheet parity of
type IIB strings and adding the Dbranes needed to ensure RRcharge conservation
(in a sense these can be viewed as twisted sector states).
Because the branes we have discussed are stable under deformations in the moduli
space, they should be mapped in a onetoone way onto each other by string dualities.
Since eleven dimensional supergravity did not appear until now in our discussion (it
does not correspond to an eﬀective weakly coupled string theory), we have to list
the relevant BPS branes of 11 dimensional supergravity. 11 dimensional supergravity
contains a three form gauge potential which can be Hodge dualized to a six form gauge
potential. Analogously to the solutions found in the previous chapter, one ﬁnds thus
a membrane (2 brane) and a ﬁve brane. 4. String dualities 140 Let us now walk once around the star in ﬁgure 4.1 in a clockwise direction and follow
the branes along this journey. Upon compactifying one of the eleven dimensions the
momentum into this direction becomes quantized. The oﬀ diagonal metric components
containing one 11 label become a Kaluza Klein gauge ﬁeld – a one form potential, which
can be Hodge dualized (with respect to the non compact directions) to a seven form
potential. The associated BPS states are zero and six branes. These become the D0
and the D6 branes in the type IIA picture. For the branes which exist already in
the uncompactiﬁed theory, there are two options within the compactiﬁcation. The
compact dimension can be transverse or longitudinal. Hence, the membrane will be
either described by a fundamental string or by a D2 brane in weakly coupled type
IIA theory, and the ﬁve brane yields the D4 brane and the NS ﬁve brane of type IIA
theory.
Compactifying further down to nine dimensions and taking the decompactiﬁcation
limit after a Tduality transformation, type IIA theory goes over into type IIB theory.
The Dbranes gain or lose one spatial direction due to the Tduality, and hence we
obtain all the Dbranes of type IIB theory. Type IIB theory possesses a symmetry
which is not depicted in ﬁgure 4.1. This is an SL (2, Z) symmetry which we do not
want to discuss in detail. For later use we state that the SL (2, Z) symmetry contains
a transformation called S duality. S duality interchanges strong with weak coupling,
the D1 brane with a fundamental string and the D5 brane with the NS ﬁve brane.
The D3 brane stays a D3 brane under Sduality.1
Type I strings are obtained by projecting out worldsheet parity in type IIB strings.
This removes the fundamental string, the NS ﬁve brane, and the D3 brane from the
spectrum of BPS states. The remaining states are the D1 and the D5 brane. Under
the strong/weak coupling duality mapping of type I theory to the SO (32) heterotic
theory, these become the fundamental string and the NS ﬁve brane of the heterotic
string. The BPS spectrum is not aﬀected when going over to the E8 × E8 heterotic
string via Tduality. The E8 × E8 theory is supposed to be dual to 11 dimensional supergravity on S 1/Z2 . Therefore, let us discuss which of the branes of 11 dimensional
supergravity survive the Z2 projection. First of all, the zero and the six branes are
projected out since the KaluzaKlein gauge ﬁeld is odd under changing the sign of the
eleventh coordinate. In order to deduce the Z2 action on the three form potential C ,
we note that the action of 11 dimensional supergravity contains a Chern Simons term
C ∧ dC ∧ dC.
This term is symmetric under the Z2 if C receives an additional sign, i.e. a C component 1
We do not include the D7 brane and its counterpart, the D instanton (related by Hodge duality),
into the discussion. 4. Dualities in Field Theory 141 containing an 11 label is even under the Z2 . Conversely, the Z2 even components of
the dual six form potential do not contain an 11 label. From the ten dimensional
perspective, a one brane and a ﬁve brane survive the Z2 projection. These are the
fundamental string and the NS ﬁve brane in the heterotic description.
Hence, we have seen that continuous changes of M theory moduli preserve the
spectrum of BPS branes. We have identiﬁed dual descriptions of branes. Note also
that not all tips of the star in ﬁgure 4.1 are connected by continuous changes of
moduli. For example, 11 dimensional supergravity on S 1 is not continuously connected
to 11 dimensional supergravity on S 1/Z2. Therefore, the BPS branes of the circle
compactiﬁed 11 dimensional supergravity have a onetoone description in type IIA
theory, but the type IIA BPS branes cannot all be given a heterotic description, and
so on. 4.2 Dualities in Field Theory Another area where supersymmetry allows insight into strongly coupled regions of
perturbatively formulated theories are supersymmetric ﬁeld theories. In this section
we will focus on four dimensional N = 1 gauge theories with matter in the fundamental
representation (supersymmetric QCD). For the various other examples we refer to the
literature (see chapter 6). In supersymmetric theories, nonrenormalization theorems
allow to study the moduli space in strongly coupled regions. In N = 1 theories, the
superpotential must be holomorphic in the ﬁelds. This often restricts its form, and the
moduli space is found by searching for ﬂat directions in the superpotential. A thorough
analysis of N = 1 SU (Nc ) gauge theory with Nf chiral multiplets in the fundamental
representation led Seiberg to the conjecture that perturbatively completely diﬀerent
looking theories are connected in moduli space. Analyzing results on beta functions
3
in such theories, one ﬁnds that for 2 Nc < Nf < 3Nc the beta function becomes
zero at a certain (strong) coupling. Hence, such gauge theories ﬂow to a conformal
ﬁxed point in the infrared (they are asymptotically free). The amazing result of
Seiberg’s analysis is that an SU (Nf − Nc ) theory with Nf chiral multiplets in the
2
fundamental representation of SU (Nf − Nc ) and Nf gauge singlets ﬂows to the same infrared ﬁxed point as the above mentioned SU (Nc ) theory. Thus, the moduli spaces of
the two theories are connected in the strong coupling region. The ﬁeld theory analysis
involves ﬁrst ﬁnding a duality map between conformal primaries at the infrared ﬁxed point and to test whether the picture is consistent under continuous deformations.
Another quite non trivial consistency check is that the ‘t Hooft anomaly matching
conditions are satisﬁed. In the present section we will sketch how the moduli spaces
of the two theories mentioned above can be connected by simply playing around with 4. Dualities in Field Theory 142 NS5 NS5 Nf D6 Nc D4 Figure 4.2: Brane setup for supersymmetric QCD. It has to be looked at in combination
with table 4.1.
NS5
NS5
D4
D6 0
–
–
–
– 1
–
–
–
– 2
–
–
–
– 3
–
–
–
– 4
–
·
·
· 5
–
·
·
· 6
·
·
–
· 7
·
·
·
– 8
·
–
·
– 9
·
–
·
– Table 4.1: Brane setup for supersymmetric QCD. The numbers in the ﬁrst line label
the dimensions. Hyphens denote longitudinal and dots transverse dimensions.
branes. Throughout this section we will neglect the back reaction of branes on the
target space geometry, i.e. we take a limit where gravity decouples.
Our ﬁrst task is to translate N = 1 SU (Nc ) supersymmetric gauge theory with
Nf chiral multiplets in the fundamental representation into a brane setup. A setup
yielding the desired theory is drawn in ﬁgure 4.2. Since it is diﬃcult to draw pictures
in ten dimensions we supplement the ﬁgure by table 4.1 where hyphens stand for
longitudinal and dots for transverse dimensions. The D4brane stretches in the sixth
direction between the two NS5 branes. Hence, its extension along the sixth dimension
is given by the ﬁnite distance of the NS5 and the NS5 brane. If this distance is
shorter than the experimental resolution, the theory on the D4branes is eﬀectively
3+1 dimensional. The positions of the NS5, NS5 and the D4 in the seventh dimension
must coincide (simply for geometrical reasons). We take Nc of such D4branes in
order to obtain SU (Nc ) gauge theory. The position of the D4branes in the transverse
directions is ﬁxed by the condition that it stretches between the NS5 and NS5 brane.
The scalar ﬁelds in the adjoint of the gauge group correspond to collective coordinates
for those positions. They are projected out by the boundary condition. Therefore, 4. Dualities in Field Theory D5
D5
F1
D3 0
–
–
–
– 143
1
–
–
·
· 2
–
–
·
· 3
–
–
·
· 4
–
·
·
· 5
–
·
·
· 6
·
·
–
· 7
·
·
·
– 8
·
–
·
– 9
·
–
·
– Table 4.2: Dual brane setup for supersymmetric QCD. This can easily be checked to
be consistent.
the theory on the D4branes can admit at most N = 1 supersymmetry (viewed from
a 3 + 1 dimensional perspective). We will argue in a moment that there is partially
unbroken supersymmetry in the above setup. Before, let us comment on the role of
the D6branes. A string starting on a D6 and ending on a D4brane transforms in
the fundamental representation of SU (Nc ). If we take Nf D6branes we obtain the
Nf desired multiplets in the fundamental representation of the gauge group. The
SU (Nf ) gauge theory becomes the ﬂavour symmetry of our supersymmetric QCD.
(Indeed, the eﬀective four dimensional coupling is obtained by integrating over the
extra dimensions. It is inversely proportional to the volume of the extra dimensions.
Since the D6brane worldvolume contains noncompact extra dimensions, the SU (Nf )
dynamics decouples and we are left with a global symmetry.)
After we successfully constructed a brane setup for the gauge theory we are interested in, we should check whether this brane setup is consistent. One could, for
example, couple it to gravity and look for explicit solutions describing such a setup.
This is, however, rather complicated. What we will do instead, is to take a setup from
which we know that it is consistent and connect it to the above setup through a chain
of string dualities. A setup of which we know that it is consistent is given in table
4.2. Here, a fundamental string (F1) stretches between two D5branes (D5 and D5 ).
This is consistent by the deﬁnition of a Dbrane. Above, we have argued that at most
N = 1 supersymmetry survives on the (interval compactiﬁed) D4brane. An argument
that this is exactly the case can be given by counting the ND directions of the setup in
table 4.2 (see also section 2.3.1.3). The D5 and the D3brane provide eight ND directions (all but the zeroth and sixth dimension). An open string starting on a D5 brane
and ending on a D3brane has four ND directions (1237). Finally, the string stretched
between the D5 and D5 brane has mixed (ND) boundary conditions along the four
dimensions: 4589. Hence, the number of ND directions is always an integer multiple
of four, indicating that the spectrum possesses some supersymmetry. (There are also
more precise methods to investigate the number of preserved supersymmetries. One
can study the conditions for vanishing gravitino and dilatino variations in the rigid 4. Dualities in Field Theory
Nf D6 144
NS5 NS5 Nf − Nc D4
Nc D4 Figure 4.3: Brane setup of ﬁgure 4.2 after pushing the D6branes past the NS5 brane.
limit, see e.g. [158] for such an analysis within the present context.) It remains to see
that the setup in table 4.2 is connected to the one which we are interested in (table
4.1). Table 4.2 contains branes of type IIB theory. Therefore, we can apply an Sduality (shortly described in the previous section) on this system. This takes the D5
and D5 brane to the NS5 and NS5 brane of table 4.1. The fundamental string (F1)
turns into a D1 string and the D3brane remains invariant under Sduality. Performing
a T duality along the ﬁrst, second and third dimension (replacing type IIB by IIA)
yields the conﬁguration of table 4.1.
In the following we will describe a path in the moduli space of the setup in ﬁgure
4.2 taking us to the dual theory found by Seiberg. We will do so by essentially
interchanging the position of the NS5 with the NS5 brane. This involves however
some subtleties which we will mention but not elaborate on. For more details we ask
the interested reader to consult[158] or literature to be given in chapter 6. Our ﬁrst
step is to move the D6branes to the left of the NS5 brane. When the D6branes cross
the NS5 brane, Nf additional D4branes stretching between the D6branes and the
NS5 brane are created[238]. After the D6branes have been moved to the left of the
NS5 brane, there is a point in moduli space where there are no D4branes stretching
between the NS5 and the NS5 brane. This can be achieved by connecting the Nc
D4branes stretching between NS5 and NS5 branes with Nc out of the Nf D4branes
which stretch between D6branes and NS5 branes. The result of performing this ﬁrst
step in moduli space is drawn in ﬁgure 4.3.
Now, the boundary conditions are such that we can displace the NS5 brane in the
seventh dimension. After doing so, it can be moved to the right of the NS5 brane 4. AdS/CFT correspondence 145
NS5 Nf D6 NS5 Nf D4
Nf − Nc D4 Figure 4.4: Brane setup for the dual gauge theory.
along the sixth dimension.2 As soon as the NS5 brane is situated to the right of the
NS5 brane, we realign it in the seventh dimension with the positions of the NS5 and
the D4branes. There are now (Nf − Nc ) D4branes starting at the D6branes passing
through the NS5 branes and ending on the NS5 brane. These we break on the NS5
brane. The picture drawn in ﬁgure 4.4 emerges.
Finally, we need to read oﬀ the perturbative formulation of the ﬁeld theory corresponding to ﬁgure 4.4. The gauge group of the theory living on the D4branes
stretching between the NS5 and the NS5 brane is SU (Nf − Nc ). There are Nf chi ral multiplets in the fundamental of the gauge group coming from strings stretching
between the Nf D4branes on the left and the Nf − Nc D4branes in the middle. The
D4branes to the left can move (ﬂuctuate) in the eighth and ninth dimension. This
2
gives rise to Nf chiral multiplets which are singlets under the gauge group.
In this section we have seen that branes can be useful tools in deriving (or at least
illustrating) quite nontrivial connections between gauge ﬁeld theories. Our purpose
was to provide the rough ideas on how this works within an example. The reader who found this interesting is strongly advised to check the literature (chapter 6) for more
details and subtleties. 4.3 AdS/CFT correspondence In this section we will describe a duality between gravity and ﬁeld theory, or from a
stringy perspective between closed string excitations and open string excitations. We
will focus on the most prominent example where the ﬁeld theory is N = 4 supercon2 Here, it is important that the NS branes do not meet when passing each other. 4. The conjecture 146 formal SU (N ) YangMills theory3 (the theory of open string excitations ending on
D3 branes) and gravity lives on an AdS5 × S 5 space (the near horizon geometry of D3
branes). In the next subsection we will state the duality conjecture and mention the
most obvious consistency checks. Instead of elaborating on the various more involved
consistency checks which have been performed in the literature, we will discuss an application of the duality. We will use the gravity side of the conjecture (the theory of
closed string excitations) to compute a Wilson loop in ﬁeld theory. This will be done
in a semiclassical approximation. We will also discuss next to leading order corrections. In order to avoid disappointment, we should mention here that we will not give
a quantitative result for the next to leading order corrections. 4.3.1 The conjecture From section 3.3 we recall that in the case of the D3 branes the truncated action in
the Einstein frame and in the string frame look almost the same. We will work in the
string frame and absorb the constant dilaton into the deﬁnition of the gravitational
coupling κ. Choosing in addition a convenient4 numerical relation between α and κ,
we can write (see (3.3.0.23))
e−C = 1 + 4πgsN α 2
,
r4 (4.3.1.1) where gs denotes the string coupling, and N = N3 is the number of D3 branes. Recall
also that the metric is (3.3.0.3) (use (3.3.0.6) and (3.3.0.7) and i, j = 0, . . . , 4, and we
parameterize the transverse space by polar coordinates, i.e. dΩ2 is the metric of a unit
5
ﬁve sphere)
C C ds2 = e 2 ηij dxi dxj + e− 2 dr2 + r2dΩ2 .
5 (4.3.1.2) Now we take the near horizon limit following the prescription
α → 0 and U ≡
r
α ﬁxed. (4.3.1.3) The ﬁrst limit ensures that the ﬁeld theory on the brane decouples from gravity living
in the bulk. The second limit implies that we zoom in on the near horizon region. It
is taken such that the mass of an open string stretching between the N D3 branes and
3 In order to distinguish between the number of branes and the number of supersymmetries we use
N for the number of supersymmetries in the present section.
4
4
The precise relation is (in Einstein frame units) κ2 = 16π7 α , where Ω5 = π3 has been used
(see (3.3.0.25)). Plugging this into (3.2.1.22), one ﬁnds that for this choice the NSNS charge of the
fundamental string equals the RR charge of the D1 string (3.3.0.13). This implies that the numerics
involved in the S duality of type IIB simpliﬁes to gs → g1 .
s 4. The conjecture 147 some probe brane at a ﬁnite distance is constant. Performing the limit (4.3.1.3) the
metric (4.3.1.2) becomes5
ds2 = α U2
√
ηij dxi dxj +
4πgs N 4πgsN dU 2
+
U2 4πgsN dΩ2 .
5 (4.3.1.4) This describes an AdS5 × S 5 geometry. Before taking a short detour on the description
of AdS5 spaces as hypersurfaces of a six dimensional space let us check the validity
region of (4.3.1.4) by focusing on the S 5 part. The radius of the S 5 is
R2 = α 4πgsN . (4.3.1.5) In order to avoid high curvature (where higher derivative corrections become important, or even the eﬀective gravity description may break down) one should take this
radius to be large, i.e.
gs N 1. (4.3.1.6) In addition we should keep the string coupling small which implies that the number
of D3 branes we look at has to be large. Now, we recall that the ﬁeld theory living on
the D3 branes is N = 4 supersymmetric SU (N ) gauge theory. The gauge coupling
2
is gY M = 2πgs (see (2.3.3.22)). So, at ﬁrst sight it seems that the gauge coupling
is small whenever the string coupling is small. However, we should also impose the
condition (4.3.1.6), in particular the large N limit. For large N SU (N ) gauge theories
‘t Hooft developed a perturbative expansion in the parameter (the ‘t Hooft coupling)
2
gY M N [435]. The condition (4.3.1.6) implies that the ‘t Hooft coupling is large whenever the eﬀective gravity description is reliable. We will argue below that gravity (or
closed type IIB strings) in the space (4.3.1.4) is dual to the gauge theory living on the D3 branes. Because of (4.3.1.6) this is a strong/weak coupling duality. One of
the ﬁrst things one should check before publishing a conjecture on dual pairs is that
the global symmetries of the dual descriptions should match. (Global symmetries are
observable.) Therefore, let us take a short detour and describe the AdS5 space as a
hypersurface in a six dimensional space. This will enable us to see the isometries of
AdS5 in much the same way as one sees the SO (6) isometry of S 5 when viewing it as
a hypersurface in six dimensional space.
The space in which we will ﬁnd an AdSp+2 space as a hypersurface is a 2 + p + 1
dimensional space with the metric
p+1
2 ds = 2
−dX−1 − 2
dX0 2
dXα. + (4.3.1.7) α=1 5
One can obtain this metric directly when dropping the requirement of an asymptotically trivial
background in the search for BPS branes in section 3.3. 4. The conjecture 148 Analogously to a sphere, the AdSp+2 space is deﬁned as the set of points satisfying
the condition
p+1
2
2
−X−1 − X0 + α=1 2
Xα = −R2 , (4.3.1.8) where R is called the radius of the AdSp+2 space. We solve this equation by
X−1 + Xp+1 = U ,
U
for i = 0, . . . , p:
X i = xi ,
R
R2
x2 U
+
,
X−1 − Xp+1 =
R2
U (4.3.1.9)
(4.3.1.10)
(4.3.1.11) where x2 = ηij xi xj and U and xi parameterize the hypersurface (4.3.1.8). Plugging
(4.3.1.9) – (4.3.1.11) into (4.3.1.7), we obtain the AdSp+2 metric
ds2 = U2
dU 2
ηij dxi dxj + R2 2 .
R2
U (4.3.1.12) Comparison with (4.3.1.4) shows us that the limit (4.3.1.3) took us to an AdS5 × S 5
space where the radii of the AdS5 and the S 5 coincide and are given by (4.3.1.5).
After this detour we can easily read of the isometries of (4.3.1.4). The isometry is
SO (4, 2) × SO (6). These isometries show up in the ﬁeld theory on the D3 branes as
follows. The SO (6) (= SU (4)) is the R symmetry of N = 4 supersymmetric YangMills theory. The beta function of the gauge theory vanishes exactly, i.e. the gauge
theory is a conformal ﬁeld theory. The SO (4, 2) part of the isometry corresponds to
the conformal group which is a symmetry in the gauge theory. Taking into account the
preserved supersymmetries6, one observes that the isometry group SO (4, 2) × SO (6)
can be extended to the superconformal group acting in the ﬁeld theory. Thus the
global symmetries of the two descriptions match. In the asymptotic region U → ∞
the AdS part of the metric (4.3.1.4) becomes (up to a conformal factor) the 3+1
dimensional Minkowski space. This is the boundary of the AdS space. The SO (4, 2)
isometry acts as the group of conformal transformations on the Minkowski space.
In this sense, one can identify the boundary of the AdS space with the location of
the D3 branes, although one should not think of the two descriptions simultaneously,
because whenever the parameters are such that the gravity description is reliable, the
perturbative description of the gauge theory breaks down and vice versa.
Moreover, one can identify the SL (2, Z) duality of the type IIB string with the
Montonen Olive duality[346, 463, 360, 457] of N = 4 super Yang Mills theory.
6 These are enhanced in the near horizon limit. 4. Wilson loop computation 149 Thus, we have seen some evidence that the AdS/CFT correspondence conjecture
holds. More checks have been performed, but we will not discuss those here. In the
following section we want to illustrate the duality by computing Wilson loops in gauge
theory using type IIB superstrings. Before doing so, let us summarize the AdS/CFT
correspondence (duality) conjecture.
• Type IIB superstrings living in an AdS5 × S 5 background are dual to open
superstrings ending on a stack of D3 branes.
• The AdS5 and the S 5 have the same radius whose value (in units of α ) is
related to the ‘t Hooft coupling of the gauge theory via equation (4.3.1.5), and
2
gY M = 2πgs .
• The type IIB string theory is in its perturbative regime if gs is small, and higher
curvature eﬀects are not dangerous as long as (4.3.1.6) holds. In this region, the
gauge theory is in the large N limit and strongly coupled.
In a somewhat weaker statement, one should replace “type IIB superstrings” by “type
IIB supergravity” and “open strings ending on D3 branes” by N = 4 SU (N ) gauge
theory. We will take the duality conjecture as stated in the items. 4.3.2
4.3.2.1 Wilson loop computation
Classical approximation A Wilson loop is the (normalized) partition function of gauge theory in the presence
of an external quark antiquark pair. A perturbative description of this situation in
a D3 brane setup for static quarks is drawn in ﬁgure 4.5. In order to employ the
AdS/CFT duality conjecture, we need to translate ﬁgure 4.5 to type IIB strings living
on AdS5 × S 5. The prescription is that the open strings in ﬁgure 4.5 translate into
a background string of type IIB theory on AdS5 × S 5. In the previous section we
have argued that the position of the N D3 branes is translated to the boundary of the
AdS5 space. (We should point out again that the emphasis is on “translated” since
the gauge theory description breaks down whenever the AdS prescription is reliable.)
Therefore, the background string should fulﬁll the boundary condition that its ends on
the AdS5 boundary are separated by a distance L. Classically, the background string
is then uniquely determined by the requirement of minimal worldsheet area. As we
will see in a moment, the picture drawn in ﬁgure 4.6 arises. The fact that the string
with minimal area goes down from the boundary into the AdS space and up again to
satisfy the boundary condition is a result of the non trivial metric. The corresponding 4. Wilson loop computation 150 N
D3 Branes L single
D3 Brane Figure 4.5: The perturbative Wilson loop setup. The quark antiquark pair corresponds to the ends of open strings on the N D3 branes. The open strings have
opposite orientation. The quark antiquark pair is chosen to be static. The dynamics
of the quarks decouples as long as the single D3 brane is very far away from the N D3
branes. The distance between the quark and the antiquark is L. Boundary
of AdS5
L Background
String Figure 4.6: The non perturbative Wilson loop setup. The quark antiquark pair
corresponds to a background string ending on the AdS5 boundary. 4. Wilson loop computation 151 calculation can be carried out explicitly. For the sake of a minor simpliﬁcation, we
2
redeﬁne the coordinate U = R u such that the metric (4.3.1.4) reads
α
ds2 = R2 u2ηij dxi dxj + du2
+ dΩ2 .
5
u2 (4.3.2.1) − det g (4.3.2.2) The worldsheet area of the background string is
S= 1
2πα dτ dσ where gαβ is the induced metric (2.1.1.2). As an ansatz for the background string we
take
X 0 = τ , X 1 = σ , X 4 = U (σ ) , (4.3.2.3) and the rest of the string positions is constant in σ and τ . The indices are assigned in
the order in which coordinates appear in (4.3.2.1), and X 4 = U (the capital U denotes
the string position in the space with the metric (4.3.2.1) and should not be confused
with the capital U in (4.3.1.3)). The ﬁrst two equations in (4.3.2.3) represent the static
gauge and the sigma dependence of U allows for the string to describe the curve of
ﬁgure 4.6. This is the simplest consistent ansatz for the given boundary conditions.
The induced metric is
ds2
ind
= −U 2 dτ 2 +
R2 U2 + (∂σ U )2
U2 dσ 2. (4.3.2.4) and thus the NambuGoto action (4.3.2.2) reads
S= T R2
2π dσ (∂σ U )2 + U 4 , (4.3.2.5) where T denotes the time interval we are considering and we have set α to one (R2
is then a dimensionless quantity giving the AdS radius in units of α ). The action
(4.3.2.5) (and also the Lagrange density L obtained by dividing S by T ) does not
depend explicitly on σ . This implies
0= ∂L
∂σ
∂L
∂L
dL
2
− ∂σ U
+
∂U
∂ (∂σ U ) dσ
∂L
(∂σ U )
−L ,
∂ (∂σ U ) = − (∂σ U )
=− d
dσ (4.3.2.6) where the EulerLagrange equations have been used in the last step. For our system
we obtain
∂σ U = ± U2
2
U0 4
U 4 − U0 , (4.3.2.7) 4. Wilson loop computation 152 where U0 is a constant related to the integration constant of the last equation in
(4.3.2.6). U0 is the lower bound on the curve in ﬁgure (4.6) at σ = 0. Thus, we can
solve for σ as a function of U
U σ=± ˜
dU U0 2
U0 ˜
U2 4
˜
U 4 − U0 , (4.3.2.8) where the plusminus sign appears due to the two branches of the curve in ﬁgure 4.6
(σ is a horizontal coordinate and U a vertical one in this ﬁgure). At the boundary
(U → ∞) the diﬀerence between the two values of σ should be L. Some straightforward
manipulations with the integral in (4.3.2.8) yield
1
L
=
B
2
4U0
where 1 B (α, β ) =
0 31
,
42 , dxxα−1 (1 − x)β −1 = (4.3.2.9)
Γ (α) Γ (β )
Γ (α + β ) denotes Euler’s Beta function. Using the identities
xΓ (x) = Γ (x + 1) , Γ (x) Γ (1 − x) = π
,Γ
sin πx 1
2 = √ π one ﬁnds for the integration constant
3 U0 = (2π ) 2
Γ 12
4 L . (4.3.2.10) This shows our earlier statement that the background string is uniquely determined
by the boundary condition. The Wilson loop W [C ] is the partition function for the
background string. For the classical approximation we ﬁnd
W [C ] = e−T E , (4.3.2.11) with
E= R2
2π dσ (∂σ U )2 + U 4. (4.3.2.12) Plugging in the classical solution (4.3.2.7) (and taking into account a factor of two due
to the two branches) yields
E= R2
π ∞
U0 dU U2
4
U 4 − U0 . (4.3.2.13) Now we split this integral into two pieces (the motivation for this will become clear
below)
E = E c + Es , (4.3.2.14) 4. Wilson loop computation 153 with
Es = R2
π ∞ dU U0 4
U 4 + U0
4
U 4 − U0 U2 , (4.3.2.15) and
Ec = − R2
π ∞ dU U0 4
U0 U2 4
U 4 − U0 . (4.3.2.16)
Let us ﬁrst discuss the integral Es . This integral is divergent due to the upper integration bound and we regularize it by a cutoﬀ Umax . The asymptotic expansion for
large Umax is
Es = = R2U0
π Umax
U0 y4 + 1 dy y2 1 R2U0
π y4 − 1
y y4 − 1 Umax
U0 =
1 R2
Umax + . . . ,
π (4.3.2.17) where the dots stand for terms going to zero as Umax is taken to inﬁnity. Thus, Es
corresponds to the self energy of the two strings in ﬁgure 4.5. It does not depend on the
distance L and diverges as the length of the string is taken to inﬁnity. Here, we observe
one interesting feature of the AdS/CFT correspondence. From the AdS perspective
Umax is a large distance, i.e. an IR cutoﬀ. On the ﬁeld theory side this appears as
a cutoﬀ for high energies, i.e. a UV cutoﬀ. This interchange between infrared and
ultraviolet cutoﬀs is a general characteristics of the correspondence[434]. Let us give a
technical remark in connection with the integral Es . Plugging in our classical solution
(4.3.2.7) into the induced metric (4.3.2.4), we ﬁnd the classical value of the induced
metric (for later use we call this R2 hαβ )
2
α
β
2
ds2
−U 2 dτ 2 +
class ≡ R hαβ dσ dσ = R U6 2
.
4 dσ
U0 (4.3.2.18) The scalar curvature computed from hαβ reads
R(2) = 2 4
U 4 + U0
.
U4 (4.3.2.19) With this information it is easy to verify that the structure of the self energy integral
is
Es = R2
4πT √
d2σ −hR(2). (4.3.2.20) (This does not contradict the Gauss–Bonnet theorem because the worldsheet of the
background string is not compact.) 4. Wilson loop computation 154 Now, let us come to the second contribution in (4.3.2.14). This will turn out to be
the more interesting one. Its computation is quite similar to the computation of U0 in
terms of L. Therefore, let us just give the result
Ec = − 4π 2
Γ 2
2gY M N
14
4 L , (4.3.2.21) 2
where (4.3.1.5) and gY M = 2πgs has been used. This is the part of the quark antiquark
potential which arises due to gluon exchange among the two quarks. It is a Coulomb potential. Since L is the only scale appearing in the setup and N = 4 supersymmetric
Yang–Mills theory has a conformal symmetry, there can be only a Coulomb potential.
Anything else would need another scale to produce an energy, but this cannot appear
due to conformal invariance.
In that respect models with less or none supersymmetry are more interesting because one can observe conﬁnement in those models. The corresponding literature is
listed in chapter 6. The case we are considering here, is the one where the AdS/CFT
correspondence is perhaps best understood. We will study a question which is interesting from a more theoretical perspective, namely whether there are corrections to
the result (4.3.2.21).
4.3.2.2 Stringy corrections Before discussing corrections to (4.3.2.21) we should envisage the possibility that
(4.3.2.21) is an exact result. There are some results which may lead to this conclusion. By analyzing the structure of possible corrections to the AdS5 × S 5 geometry,
physicists[43, 340, 276] found that this geometry is exact. Still, there is a very simple
argument destroying the hope that (4.3.2.21) might be exact. Namely, the above Wilson loop computation can also be performed in the perturbative regime, where the ‘t
Hooft coupling is small. Then one ﬁnds, of course, also a Coulomb law but the dependence on the ‘t Hooft coupling is linear instead of a square root dependence (which
actually cannot be obtained in a perturbative calculation). This does not contradict
the result (4.3.2.21) but tells us that taking the ‘t Hooft coupling smaller should result
in corrections such that ﬁnally for very small ‘t Hooft coupling the square root like
dependence goes over into a linear one.
After we have excluded corrections to the AdS5 × S 5 geometry, we will study
ﬂuctuations of the IIB string around the background string in ﬁgure 4.6. That is, we
consider the Wilson loop as the quantum mechanical partition function
W [C ] = [DδX ] [Dδθ] e−SIIB (X +δX,δθ), (4.3.2.22) 4. Wilson loop computation 155 where δX denote bosonic ﬂuctuations and δθ fermionic ones (the fermionic background
of the string is trivial). Before going into the details of the computation, let us describe
the expansion we are going to perform. From (4.3.2.2) and (4.3.2.1) we see that the
square root of the ‘t Hooft coupling appears as an overall constant in front of the metric.
(This is also true for terms containing fermions.) Therefore, the expansion parameter
(or α in section 2.1.3) is identiﬁed with the inverse square root of the ‘t Hooft coupling.
The expression (4.3.2.22) can be computed as a power series in this parameter. In
particular, the next to leading order correction to (4.3.2.21) will not depend on the ‘t
Hooft coupling. It is this correction we will discuss in some detail in the following. In
order to be able to use (4.3.2.22) for explicit calculations we need to know the type
IIB string action in an AdS5 × S 5 background. Fortunately, this has been constructed
in the literature[340]. These authors gave a type IIB action in the Green Schwarz
formalism, which is appropriate in the presence of nontrivial RR backgrounds. The
construction is similar to the one discussed in section 2.1.1.3. One uses target space
supersymmetry and kappa symmetry as a guide. The technicalities are rather involved
and we will not discuss them here. Because we will restrict ourselves to terms second
order in ﬂuctuations we only need a truncated version of the action of type IIB strings
on AdS5 × S 5. The complete action does not contain terms with an odd number of target space fermions, in particular no terms linear in target space fermions. Since the
fermionic background is trivial, contributions quadratic in ﬂuctuations can either have
two fermionic ﬂuctuations and no bosonic ﬂuctuation or only bosonic ﬂuctuations.
The part of the action which is quadratic in the ﬂuctuations consists of a sum of terms
with only bosonic ﬂuctuations and terms with only fermionic ﬂuctuations.
Let us discuss the bosonic ﬂuctuations ﬁrst. The type IIB action is a kappa
symmetric extension of (4.3.2.2). For the bosonic ﬂuctuations only the contribution
(4.3.2.2) is relevant (for the lowest non trivial contribution). As in section 2.1.3 we parameterize the ﬂuctuations by tangent vectors to geodesics connecting the background
with the actual value, i.e. we perform a normal coordinate expansion. The quantum
ﬁelds are
a
ξ a = Eµ ξ µ , (4.3.2.23) a
where Eµ are the vielbein components obtained by taking the square root of the
diagonal metric components. A number as a label on a ξ will stand for a ﬂat index
(a), unless stated explicitly otherwise. The computation of the term second order in the ξ a is a bit lengthy but straightforward with the information given in section 2.1.3.
(The only diﬀerence to section 2.1.3 is that we expand now a NambuGoto action
instead of a Polyakov action.) Before giving the result it is useful to perform a local 4. Wilson loop computation 156
x4 = u Background
String
ξ µ=4 γ
ξ µ=1 x1 = σ Figure 4.7: Perpendicular and longitudinal ﬂuctuations in the one–four plane.
Lorentz rotation in the space spanned by the ξ a . The rotation is7
ξ
ξ⊥ = cos α sin α
− sin α cos α ξ1
ξ4 , (4.3.2.24) with
cos α = 2
U0
, sin α =
U2 4
U 4 − U0
.
U2 (4.3.2.25) Note that the determinant of the matrix appearing in (4.3.2.24) is one. The ﬁelds ξ
and ξ ⊥ describe ﬂuctuations parallel and perpendicular to the worldsheet, respectively.
This is illustrated in ﬁgure 4.7. Fluctuations drawn into ﬁgure 4.7 carry Einstein indices which we indicated explicitly. The angle γ is given by the slope of the background
string
tan γ = ∂σ U. (4.3.2.26) ξ µ=4 − tan γ ξ µ=1 (4.3.2.27) The combination vanishes for
tan γ =
7 ξ µ=4
,
ξ µ=1 We employ the symmetry of the background string and take 0 ≤ σ ≤
(4.3.2.7). (4.3.2.28)
L
2 and the plus sign in 4. Wilson loop computation 157 i.e. if ξ µ=1 + ξ µ=4 is tangent to the background string. Thus, the combination in
(4.3.2.27) is normal to the background string. Transforming the indices to ﬂat ones
(with the vielbein) and orthonormalization yields ξ and ξ ⊥ with the given interpretation. When writing down the Lagrangian second order in the ﬂuctuations, we can
set R = 1 since we know the general R dependence (viz. none) from the argument
given above. The Lagrangian for the bosonic ﬂuctuations comes out to be
(2) (2) (2) Lbosons = LAdS5 + LS 5 (4.3.2.29) with
(2) LAdS5 = √
1
−h 2 a=2,3,⊥ + R(2) − 4
(2) LS 5 = 1√
−h
2 ξ a ∆ξ a − 2 ξ 2 ξ⊥ 2 2 − 2 ξ3 2 , (4.3.2.30) 9 ξ a ∆ξ a , (4.3.2.31) a =5 where total derivative terms have been dropped (the ﬂuctuations should satisfy Dirichlet boundary conditions in order not to change the classical boundary conditions).
Further, ∆ denotes the two dimensional Laplacian with respect to the metric hαβ
(4.3.2.18) and R(2) is the corresponding scalar curvature (4.3.2.19). We observe that
the longitudinal ﬂuctuations ξ 0 and ξ drop out of the action. Hence, we can ﬁx the
worldsheet diﬀeomorphisms via
ξ 0 = ξ = 0. (4.3.2.32) If the normalization of the functional integral in (4.3.2.22) contains a division by the
volume of the worldsheet diﬀeomorphisms, we cancel the ξ 0 and ξ integration against
this term in the normalization. This may be problematic, and we will comment on
this issue later.
It remains to study the fermionic ﬂuctuations. Since the fermionic background is
trivial we just need to copy the Lagrangian from[340] (truncated to quadratic terms)
and to plug in our background. The result of the copying task is
LF 1√
a
ˆ
a
ˆ
¯ ˆa
¯ ˆa
−hhαβ Eα − iθI γ ˆ (Dαθ)I Eβ − iθJ γ ˆ (Dβ θ)J
2
a
ˆ¯a
¯ ˆa
ˆ
−i αβ Eα θ1 γ ˆ (Dβ θ)1 − θ2 γ ˆ (Dβ )2 . =− (4.3.2.33) First, we need to explain some of the notation. The index a = 0, . . . , 9 labels the
ˆ
tangent space coordinates of AdS5 × S 5. Later, we will use an index a = 0, . . . , 4 for 4. Wilson loop computation 158 the tangent space of AdS5 and a = 5, . . . , 9 for the tangent space of S 5. The vielbein
with a worldsheet index is
a
ˆ
a
ˆ
Eα = Eµ ∂αX µ , (4.3.2.34) where X µ is the position of the background string. The indices I, J = 1, 2 label the
two target space supersymmetries. The derivative Dα is deﬁned as
(Dαθ)I = δ IJ ≡ αθ ∂α +
I − i
2 1
i
ab
(∂α X µ) ωµ γ ab −
4
2 IJ IJ a
(∂α X µ) Eµ γ a θJ a
(∂α X µ ) Eµ γ aθJ . (4.3.2.35) ab
Here, ωµ denotes the target space spin connection and we have used the fact that our background is trivial in S 5 directions. The gamma matrices γ ˆ = γ a, iγ a
ˆa satisfy SO (4, 1) and SO (5) Cliﬀord algebras, respectively. The θI are sixteen component
spinors each. They are conveniently labeled by a double spinor index θαα where α is
a spinor index in the tangent space of the AdS5, and α a spinor index in the tangent
space of S 5. The γ a and γ a are four times four matrices tensored with four times
four identity matrices. (In the following we will suppress target space spinor indices
in order to avoid confusion with worldsheet indices which are also labeled by small
Greeks.)
We do not intend to give a derivation of (4.3.2.33) but let us have a brief look
at its structure before proceeding. Expression (4.3.2.35) is a tensor (density) with
indices αβ contracted either with hαβ or αβ . The terms with hαβ can be thought
of as arising from the replacement (2.1.1.36)8 whereas the αβ contracted terms come
from the Wess Zumino term (2.1.1.38) needed for kappa symmetry. The details diﬀer
from the discussion in section 2.1.1.3 due to the diﬀerent target space geometry and
the RR four form ﬂux.
For our background, the Lagrangian (4.3.2.33) can be written in a compact way
√
¯¯
LF = − −h θ1 , θ2 αβ a
2iEµ (∂α X µ ) γ aP− β
1−B
αβ
a
−1 − B
2iEµ (∂α X µ) γ aP+ β θ1
, (4.3.2.36)
θ2 where X µ stands for the background position of the string and
αβ
P± = B=
8 1
2 αβ hαβ ± √
,
−h
1
αβ a b
√
Eµ Eν (∂α X µ ) (∂β X ν ) γ ab.
2 −h (4.3.2.37)
(4.3.2.38) In general, the combination in (2.1.1.36) contains also terms higher than quadratic order in the
fermions. This is because the superalgebra is altered in the AdS5 × S 5 case as compared to a ﬂat
target space. This should be clear by noting that the isometries form a subgroup of the supersymmetry
transformations. 4. Wilson loop computation 159 As usual, a gamma with a multiple index is the antisymmetrised product of gamma
matrices.
It is useful to perform the rotation (4.3.2.24) also on the spinors (with α as given
in (4.3.2.25))
θI = cos α
α
− sin γ 14 ψ I .
2
2 (4.3.2.39) In order to compute the partition function, we should ﬁx the kappa symmetry. This
is conveniently done in terms of the nilpotent matrices
γ± = 10
γ ± γ1 .
2 (4.3.2.40) In analogy to section 2.1.1.3, we choose the kappa ﬁxing conditions
γ − ψ 1 = 0 , γ + ψ 2 = 0. (4.3.2.41) We assume that the integration over spinors not satisfying (4.3.2.41) cancels the volume
of kappa transformations appearing as a normalization factor in the functional integral.
This may be problematic, and we will comment on this issue later. Spinors satisfying
the kappa ﬁxing condition are then governed by the Lagrangian
√
¯¯
LF = − −h ψ 1, ψ 2
where ± iγ + +
2
−
−2
iγ ψ1
ψ2 − , (4.3.2.42) are tangent space derivatives deﬁned as follows
± = eτ
0 τ ± eσ
1 σ = 1
U τ ± 2
U0
U3 σ, (4.3.2.43) where eτ and eσ are two dimensional (inverse) vielbein components obtained from the
0
1
square roots of the diagonal elements of hαβ . Note also that the covariant derivative
simpliﬁes when acting on spinors satisfying (4.3.2.41). Deﬁning partial tangent space
derivatives analogously to (4.3.2.43) one ﬁnds
±ψ I = ∂± ± ω±
ψI ,
2 (4.3.2.44) where
01
01
ω± = eτ ωτ ± eσ ωσ
0
1 (4.3.2.45) 01
are tangent space components of the two dimensional spin connection ωα computed
from the zweibeinen deﬁned in (4.3.2.43). Let us further deﬁne the matrices ρ+ = 00
γ0 0 , ρ− = 0 −γ 0
0
0 . (4.3.2.46) 4. Wilson loop computation 160 These are the same matrices as in (2.1.1.17) with i replaced by γ 0/2. Finally, we
rewrite (4.3.2.42) in a suggestive way as follows
LF √
¯
¯
= − −h ψ2, −ψ1
√
¯
¯
= − h ψ2 , −ψ1 iγ − −2 −iγ + + 2iγ 0 −2 −2iγ √
¯
¯
= 2 −h ψ2, −ψ1 (iρm 0 m −2 + −2 + 1) ψ1
ψ2 ψ1 − ψ2
ψ1 − ψ2
, (4.3.2.47) where in the second line (4.3.2.41) has been used and a repeated index m stands
for the sum over the labels + and −. Comparison with the expressions in section
2.1.1.2 shows that the part of the action containing target space spinor ﬂuctuations
‘metamorphosed’ into an action for worldsheet spinors after imposing the kappa ﬁxing
condition (4.3.2.41). The diﬀerence is that the derivative contains the spin connection
due to the non trivial worldsheet metric hαβ (4.3.2.18), and the mass terms appearing
due to the constant non vanishing curvature of the AdS space.
Now we have collected all the information needed to express the second order ﬂuctuation contribution to (4.3.2.22) in terms of determinants of two dimensional diﬀerential
√
operators. (For Dirac operators one uses the formal identity det A = det A2 .)
Integration over the ﬂuctuations leads to determinants of operators which can be
read oﬀ from (4.3.2.30), (4.3.2.31) and (4.3.2.47). The corrected expression for the
Wilson loop reads
W [C ] = e−T Eclass 1
det −∆F − 4 R(2) + 1
1 5 det (−∆ + 2) det 2 −∆ + 4 − R(2) det 2 (−∆) . (4.3.2.48) The exponential is the classical contribution with Eclass given by (4.3.2.14), (4.3.2.20)
and (4.3.2.21). Note also that the operator appearing in the numerator of (4.3.2.48) is a
four times four matrix. The Laplacian acting on worldsheet fermions F is η mn n m .
Unfortunately it is not known how to evaluate the determinants in (4.3.2.48) exactly.
What is known exactly are the divergent contributions. These are given in (2.1.3.33).
They are of the form
Ediv ∼ √
d2σ −hR(2). (4.3.2.49) Comparing with (4.3.2.20), we ﬁnd that this divergence renormalizes the selfenergy
which is inﬁnite anyway. A correction to the Coulomb charge of the quarks will be
ﬁnite. Unfortunately, we cannot give it in a more explicit way (as a number).
In addition, there is also a conceptual puzzle with the divergent contribution.
Although for our problem it is not relevant, it should not be there. The argument 4. Strings at a TeV 161 that something might have gone wrong goes as follows. The string action is equivalent
to a Polyakov type action, at least at a classical level (see section (2.1.1.1)). The
Polyakov action is conformally invariant, and in a consistent string background the
conformal invariance should not be broken by quantum eﬀects. Therefore, divergences
which introduce a cutoﬀ (or renormalization group scale) cannot occur. Indeed, it
was argued in[149] that a treatment analogous to ours but with a Polyakov instead
of the NambuGoto action leads to a ﬁnite result. This treatment is a bit more
complicated since the worldsheet metric appears as an independent ﬁeld which also
ﬂuctuates. The advantage is however that subtle contributions due to the occurring
integral measures are well understood. Such contributions are typically of the structure
(4.3.2.49)[16, 193, 328]. (Note, however, that if the worldsheet metric is identiﬁed with
the induced metric, the term in (4.3.2.49) is not really distinguishable from (∂X )2
terms (see e.g.[155])). In our derivation, we have mentioned already two places where
nontrivial measure contributions could arise. This could happen when we cancel the
integration over the longitudinal ﬂuctuations against the volume of the worldsheet
diﬀeomorphisms and in the kappa ﬁxing procedure. Unfortunately, the NambuGoto
case is less understood than the Polyakov formulation. (For a recent attempt to ﬁx
the functional measures in the bosonic part see[347].) Fortunately, the result of the
better understood calculation in the Polyakov approach is identical to the one given
here (up to the irrelevant divergence)[149].
With these open questions we close our discussion on the AdS/CFT correspondence. The reader who wants to know more will ﬁnd some references in chapter 6. 4.4 Strings at a TeV So far we have not determined the numerical value of the string scale (set by α ) in
terms of a number. We restricted our discussions mostly to the massless excitations
of the string. This was motivated by the belief that the string scale (in energies)
is large compared to observed energy scales. Often it is comparable to the Planck
scale. This identiﬁcation is motivated by studies of heterotic weakly coupled strings
which provided for a long time the most promising starting point in constructing
phenomenologically interesting models. Such models are obtained by compactifying
the ten dimensional heterotic (mostly E8 × E8) string down to four dimensions on a
Calabi–Yau manifold. Let us give a rough estimate for the resulting four dimensional
couplings. The eﬀective four dimensional heterotic action is of the form
Shet = d4x V −8 (4)
−
− lh 6trF 2 + . . . ,
2lR
gh h (4.4.0.1) 4. Strings at a TeV 162 where we drop details which are not relevant for the present estimate on scales. In
(4.4.0.1) lh is the heterotic string scale (set by α ), gh the heterotic string coupling
(ﬁxed by the dilaton vev), and V is the volume of the compact space. The quantities
in which four dimensional physics is usually described are the four dimensional Planck
mass Mp and the gauge coupling gY M . These are related to the input data (gh , lh and
V ) as follows
2
Mp = V
28 ,
gh lh 1
2
gY M = V
2 6.
gh l h (4.4.0.2) Expressing gh in the ﬁrst equation in terms of the second equation and further deﬁning
the string mass scale as Mh = 1/lh the above equation can be rewritten as
√
V
Mh = gY M Mp , gh = gY M 3 .
lh (4.4.0.3) Now we assume that a gY M ∼ 0.2 is a realistic value. (This is the gauge coupling of the minimal supersymmetric standard model at the GUT scale.) Plugging gY M = 0.2
into the ﬁrst equation in (4.4.0.3) we ﬁnd that the heterotic string scale is
Mh ∼ 1018GeV, (4.4.0.4) i.e. of the order of the Planck scale. The second equation in (4.4.0.3) implies that the
compact space is also of the Planck size if we want to stay within the region where the
string coupling is small.
Now let us investigate how the above estimates on scales are altered in a theory containing branes. Phenomenologically interesting models arise also as orientifold
compactiﬁcations of type II theories. As we have seen in section 2.4, these contain typically Dbranes on which the gauge interactions are localized whereas the gravitational
sector corresponds to closed string excitations which propagate in all dimensions. Assuming that the gauge sector (and charged matter) is conﬁned to live on Dpbranes
the eﬀective action for the orientifold compactiﬁcation will be of the form
Sori = d10x 1
2 8 R−
gII lII dp+1x 1 p− trF
gII lII 3 2 , (4.4.0.5) where lII and gII are the string scale and coupling of the underlying type II theory,
respectively. Assuming further that our orientifold construction is such that the compact space has dimensions which are transverse to all relevant Dbranes we denote by
V⊥ the volume of the compact space transverse to the branes and by V the volume of
the compact space longitudinal to the branes (such that the overall compact volume
is V = V⊥ V ). With this notation the four dimensional action reads
Sori = d4 x V V⊥
2 8 R−
gII lII d4 x V
p−
gII lII 3 trF 2 , (4.4.0.6) 4. Strings at a TeV 163 from which we obtain the four dimensional Planck length lp and gauge coupling gY M
V V⊥
1
= 28
2
lp
gII lII , 1
2
gY M = V
p−
gII lII 3 . (4.4.0.7) Hence, the four dimensional Planck mass (Mp = 1/lp) and the string coupling gII are
2
Mp = −
v⊥ lII2 v (gY M )4 2
, gII = gY M v , (4.4.0.8) where
3−
p−
v = V lII p , v⊥ = V⊥ lII 9 (4.4.0.9) are dimensionless numbers describing the size of the compact space in string scale
units. The relations (4.4.0.8) allow to take the string length lII larger than the four
dimensional Planck length lp. This can be achieved by taking v⊥ large. The size of the
parallel volume is taken to be of the “string size”, i.e. v ∼ 1. If the parallel volume is
smaller than the string size, we Tdualize with respect to the smaller dimension. This
dimension will then contribute to the perpendicular volume since the string changes
boundary conditions. Hence, the v < 1 case is T dual to the considered case of large
v⊥ . On the other hand if v > 1, the second equation in (4.4.0.8) tells us that in this case the string coupling becomes strong, and our description breaks down. (Moreover,
it is problematic for gauge interactions to be compactiﬁed on large volumes because
the corrections to the four dimensional gauge interactions are usually ruled out by
experimental accuracy.)
Let us analyse in some detail what happens if we choose a TeV for the string scale.
This is about the lowest value which is just in agreement with experiments. (For a
lower value massive string excitations should have shown up in collider experiments.)
With Mp ∼ 1016TeV we ﬁnd
v⊥ ∼ 1028 → V⊥ = 1028 1
.
(TeV)9−p (4.4.0.10) The Planck length is about 10−33cm and hence in our units one TeV corresponds to
1/ 10−18 m . Thus we obtain
V⊥ ∼ 1028−(9−p)18 (m)9−p . (4.4.0.11) For the case p = 8 (one extra large dimension) we obtain that the perpendicular
dimension is compactiﬁed on a circle of the size
p = 8 → R⊥ ∼ 1010km. (4.4.0.12) 4. Strings at a TeV 164 Such a value is certainly excluded by observations. (In the next subsection we will
compute corrections to Newton’s law due to KaluzaKlein massive gravitons and see
that the size of the compact space should be less than a mm.) For p = 7 we obtain
(distributing the perpendicular volume equally on the two (extra large) dimensions)
p = 7 → R⊥ ∼ 0.1mm. (4.4.0.13) This value is just at the edge of being experimentally excluded. The situation improves
the more extra large dimensions there are. For example in the case p = 3 (and again
a uniform distribution of the perpendicular volume on the six dimensions (V⊥ = R6 ))
⊥
we obtain p = 3 → R⊥ ∼ 10−10m, (4.4.0.14) which is in good agreement with the experimental value (Rexp = 0...0.1mm).
⊥
We have seen that D branes allow the construction of models where the string
scale is as low as a TeV. (Note also, that in the above discussion we can perform
Tdualities along the string sized parallel dimensions. This changes p but leaves the
large extra dimensions unchanged. Actually, it might be preferable to have p = 3 in
order to avoid Kaluza–Klein gauge bosons of a TeV mass.) This gives the exciting
perspective that string theory might be at the horizon of experimental discovery. In
near future collider experiments, massive string modes would be visible. In addition,
the extra large dimensions could be also discovered soon. This can happen either by
the production of KaluzaKlein gravitons in particle collisions or by short distance
Cavendish like experiments. However, it might as well be the case that models with
less “near future discovery potential” are realized in nature.
Apart from the prospect of being observed soon, strings at a TeV scale are interesting for another reason. If the string scale is at a TeV, we would call this a fundamental
scale. Thus the hierarchy problem would be rephrased. With the fundamental scale at
a TeV we should wonder why the (four dimensional) Planck scale is so much higher, or
why gravitational interactions are so much weaker than the other known interactions.
This hierarchy is now attributed to the size of the extra large dimensions. Supersymmetry may not be necessary to explain the hierarchy between the Planck scale
and the weak scale. Therefore, in the above models supersymmetry could be broken
already by the compactiﬁcation. In such models the question of stability is typically
a problematic issue.
The above considerations are also interesting if one does not insist on a direct
connection to string theory. If one just starts ‘by hand’ with a higher dimensional
setup containing branes, one would also obtain the ﬁrst equation in (4.4.0.8). In this 4. Corrections to Newton’s law 165 case, one calls lII the higher dimensional Planck length, which in turn can be chosen
to be 1/TeV. 4.4.1 Corrections to Newton’s law In the previous section we stated that observations provide experimental bounds on
the size of extra dimensions. In the brane setup in which we found the possibility of
large (as compared to the Planck length) extra dimensions, these extra dimensions
are typically tested only by gravitational interactions. Therefore, let us describe the
inﬂuence of additional dimensions on the gravitational interaction in some more detail.
We will be interested in the Newtonian limit of gravity. For simplicity, we assume that
the space is of the structure M4 × T n , where M4 is the 3 + 1 dimensional compact
space and T n is an n dimensional torus of large volume. (There might be an additional
compact space of Planck size. This does not enter the computation carried out below.)
The analysis we will carry out here is similar to the discussion of the massless scalar
in section 2.1.5.1, where the role of the scalar is taken over by the Newton potential.
Let us arrange the spatial coordinates into a vector (x, y), where x corresponds to
the M 4 and y to the T n . For simplicity we assume that the torus is described by a
quadratic lattice and the uniform length of a cycle is 2πR, i.e.
y ≡ y + 2πR. (4.4.1.15) The n + 4 dimensional Newton potential Vn+4 of a point particle with mass µ located
at the origin is given by the equation
∆n+3 Vn+4 = (n + 1) Ωn+2 Gn+4 µ δ (n+3) (x, y) , (4.4.1.16) where ∆n+3 is the three dimensional ﬂat Laplacian and Ωn+2 is the volume of a unit
n + 2 sphere. Any solution to (4.4.1.16) should be periodic under (4.4.1.15). This can
be ensured by expanding the potential in terms of eigenfunctions ψ k (y) of a Laplace
operator. The eigenvalue equation is
∆n ψk (y) = −m2 ψk (y) .
k (4.4.1.17) Thus an orthonormal set of eigenfunctions is
ψk = 1
(2πR) ei R y,
k n
2 (4.4.1.18) where k is an n dimensional vector with integer entries. We expand the higher dimensional Newton potential into a series of the eigenfunctions with r = x dependent
coeﬃcients
Vn+4 = φk (r) ψk (y) .
k (4.4.1.19) 4. Corrections to Newton’s law 166 Plugging this ansatz into equation (4.4.1.16) determines the Fourier coeﬃcients9
φk (r) = − Ωn Gn+4 µ ψk (0) 1 − k
e R.
2
r (4.4.1.20) Now, we consider the case that all particles with which we can test the gravitational
potential are localized at y = 0. (This is natural from the brane picture since we can
test gravity only with matter which is conﬁned to live on the brane. Recall that we
neglected the eﬀects of the Planck sized longitudinal compact dimensions.) We are
interested in the Newton potential at y = 0. This comes out to be
V4 ≡ Vn+4 = − G4 µ
r  k e−r R , (4.4.1.21) k where the four dimensional and the higher dimensional Newton constant are related
via
G4 = Ωn Gn+4
.
2 (2πR)n (4.4.1.22) For k = 0 we obtain the usual four dimensional Newton potential. The other terms
are additive Yukawa potentials. They arise due to the exchange of massive Kaluza
Klein gravitons.
Experimentalists usually parameterize deviations from Newton’s law via the expression [323]
V4 (r) = − r
G4 µ
1 + αe− λ .
r (4.4.1.23) In the paper[323] the experimental values are discussed. These maybe outdated by
now but for us only the order of magnitude is important (and the fact that so far no
deviation from Newton’s law has been observed). Depending on the size of α an upper
bound on λ varying from the µm range to the cm range has been measured. This tells
us that a scenario with two extra large dimensions is almost excluded whereas setups
with more than two extra large dimensions are in agreement with the experimental
tests of Newton’s law. 9 Here, one uses the completeness relation satisﬁed by the ψk . Chapter 5 Brane world setups
In the last section of the previous chapter we have argued that branes allow for scenarios with large extra dimensions transverse to the brane. This is because those extra
large dimensions can be tested only via gravitational interactions which are (due to
their weakness) measured only at scales down to about 0.1 mm. We obtained such
models via investigations of string theory. One could, however, just postulate the existence of branes (on which charged interactions are located). In this last chapter we will
take this latter point of view and not worry whether the setups we are going to discuss
have a stringy origin. Because in the presence of branes we can attribute the hierarchy
between the Planck and the weak scale to the size of the transverse dimensions, we
do not need supersymmetry in such setups. Without supersymmetry, quantum eﬀects
usually create vacuum energies. A non vanishing vacuum energy on a brane will back
react on the geometry of the space in which the brane lives. Taking into account such
back reactions leads to so called warped compactiﬁcations. This means that the higher
dimensional geometry is sensitive to the position of a brane. The most prominent example of such warped compactiﬁcations are the Randall Sundrum models which we
will discuss next. 5.1
5.1.1 The Randall Sundrum models
The RS1 model with two branes In the model we are going to describe in this section there is one extra dimension which
will be denoted by φ. The ﬁve dimensional space is a foliation with four dimensional
Minkowski slices. The ﬁfth dimension is compactiﬁed on an orbifold S 1/Z2 . 3branes
are located at the orbifold ﬁxed planes (at φ = 0 and φ = π ) . Hence the action is of 167 5. The RS1 model with two branes 168 the form
S = Sbulk + Sb1 + Sb2 , (5.1.1.1) where Sb1 and Sb2 denote the actions on the branes. For the bulk action we take ﬁve
dimensional gravity with a bulk cosmological constant,
π
√
dφ −G 2M 3 R − Λ ,
Sbulk = d4x
(5.1.1.2)
−π where M denotes the ﬁve dimensional Planck mass and GM N is the ﬁve dimensional
metric. The branes are located in φ and we identify the brane coordinates with the
remaining 5d coordinates xµ , µ = 0, . . . , 3. Then the induced metrics on the branes
are simply
b1
b2
gµν = Gµν φ=0 , gµν = Gµν φ=π . (5.1.1.3) We assume that ﬁelds being localized on the brane are in the trivial vacuum and
take into account only nonzero vacuum energies on the branes. Calling those vacuum
energies T1 and T2 , the brane actions read
Sb1 + Sb2 = − d4x T1 −g b1 + T2 −g b2 , (5.1.1.4) where the ﬁrst (second) term on the lhs is attributed to the ﬁrst (second) term on
the rhs. Instead of working out the solutions to the system on an interval S 1/Z2 it
is technically easier to construct a solution in a non compact space, such that the
solution is periodic in
φ ≡ φ + 2π, (5.1.1.5) φ → −φ. (5.1.1.6) and even under A vacuum with this property yields then automatically a compact interval in φ. (The
equivalent1 and more complicated alternative is to deﬁne the theory on an interval from
the very beginning and take into account surface terms when deriving the equations
of motion as well as Gibbons Hawking[200] boundary terms (for a discussion in the
context of brane worlds see also[132, 133]).) With these remarks the Einstein equations
of motion read (capital indices run over all dimensions M, N = 0, . . . , 4)
√
1
−G RM N − GM N
2
−
1 = √
1
Λ −GGM N +
3
4M 2 Ti
i=1 bi µ ν
−g bi gµν δM δN δ (φ − φi ) , I acknowledge discussions with Radoslaw Matyszkiewicz on this topic. (5.1.1.7) 5. The RS1 model with two branes 169
φ φ
−π −2π 0 π Figure 5.1: The periodic modulus function.
with φ1 = 0 and φ2 = π . The delta functions appearing on the rhs of (5.1.1.7) are
deﬁned on a real line. The most general metric ansatz possessing a four dimensional
Poincar´ transformation as isometry is
e
2
ds2 = e−2σ(φ) ηµν dxµ dxν + rc dφ2. (5.1.1.8) We could rescale φ such that the rc dependence drops out, but that would change
the periodicity condition (5.1.1.5). Plugging this ansatz into the equations of motion
(5.1.1.7) yields (a prime denotes diﬀerentiation with respect to φ)
6σ 2
2
rc
3σ
2
rc Λ
,
4M 3
T2
T1
δ (φ) +
δ (φ − π ) .
4M 3rc
4M 3rc =−
= (5.1.1.9)
(5.1.1.10) The solution to (5.1.1.9) is
σ = rc φ −Λ
,
24M 3 (5.1.1.11) where the modulus function is deﬁned as usual in the interval −π < φ < π ,
φ = −φ
φ , −π < φ < 0
.
, 0<φ<π (5.1.1.12) This ensures that the solution is even under φ → −φ. In order to incorporate
(5.1.1.5), we deﬁne the modulus function on the real line by the periodic continuation of (5.1.1.12). The resulting function is drawn in ﬁgure 5.1. Away from the
points at φ = 0 and integer multiples of π , the second derivative of σ vanishes and 5. The RS1 model with two branes 170 (5.1.1.10) is fulﬁlled in those regions. In order to take into account the delta function
sources in (5.1.1.10), one integrates this equation over an inﬁnitesimal neighborhood
around the location of the brane sources. This gives rise to the constraints
T1 = −T2 = 24M 3k , with k2 = − Λ
24M 3 (5.1.1.13) on the parameters of the model. These constraints can be thought of as ﬁne tuning
conditions for a vanishing eﬀective cosmological constant in four dimensions. We will
come back to this point in section 5.2.3. Our ﬁnal solution is
2
ds2 = e−2krc φ ηµν + rc dφ2, (5.1.1.14) where k2 is deﬁned in (5.1.1.13), and we take k to be positive (for a negative k just
redeﬁne φ → π − φ). We observe that by taking into account the back reaction of the branes onto the
geometry, we obtain a metric which depends on the position in the compact direction.
For the particular model we consider this dependence is exponential. That opens up
an interesting alternative explanation for the large hierarchy between the Planck scale
and the weak scale. We take all the input scales (M , Λ , rc ) to be of the order of the
Planck scale. First, we should check whether this provides the correct four dimensional
Planck mass. To this end, we expand a general 4d metric around the classical solution
2
ds2 = e−2krc (ηµν + hµν ) dxµ dxν + rc dφ2 . (5.1.1.15) 2
In principle we should also allow the fourfour component of the metric rc to ﬂuctuate.
Since rc is an integration constant, such ﬂuctuations will be seen as massless scalars in the eﬀective four dimensional theory. This is a common problem known as moduli
stabilization problem. We will assume here that some unknown mechanism gives a
2
mass to the ﬂuctuations of G44 and take it to be frozen at the classical value rc . The
KaluzaKlein gauge ﬁelds Gµ4 are projected out by the Z2 . Plugging (5.1.1.15) into the
action and integrating over φ yields the eﬀective action for four dimensional gravity
2
Sef f = Mp √
d4 x −gR(4) (g ) , (5.1.1.16) where R(4) (g ) denotes the four dimensional scalar curvature computed from gµν =
ηµν + hµν and the four dimensional Planck mass Mp is given by
2
Mp = M 3 rc π
−π dφe−2krc φ = M3
1 − e−2krc π .
k (5.1.1.17) This tells us that choosing ﬁve dimensional scales of the order of the Planck scale gives
the correct order of magnitude for the four dimensional Planck scale. 5. The RS1 model with two branes 171 Now, let us consider matter living on the branes. On the ﬁrst brane located at
φ = 0, the induced metric is just the Minkowski metric and Lagrangians for matter
living on that brane will just have their usual form. On the other hand, matter living
on the second brane (located at φ = π ) feels the φ dependence of the bulk metric. Let
us focus on a Higgs ﬁeld being located at the second brane. Its action will be of the
form
b2
SHiggs = d4xe−4krc π 2
e2krc π η µν Dµ H †Dν H − λ H 2 − vo 2 , (5.1.1.18) where the overall exponential factor originates from the determinant of the induced
metric. Rescaling the Higgs ﬁeld H such that the kinetic term in (5.1.1.18) takes its
canonical form induces the rescaling
v0 → vef f = e−krc π v0 . (5.1.1.19) This means that a symmetry breaking scale which is written as v0 into the model
eﬀectively is multiplied by a factor of e−krc π . Repeating the above argument for any
massive ﬁeld, one ﬁnds that any mass receives such a factor
m0 → mef f = e−krc π m0 , (5.1.1.20) when going to an eﬀective description in which kinetic terms are canonically normalized. Choosing krc ≈ 10 (which is roughly a number of order one), one can achieve that the exponential in (5.1.1.20) takes Planck sized input masses to eﬀective masses
of the order of a TeV. Hence, in the above model we can obtain the TeV scale from
the Planck scale without introducing large numbers, provided we live on the second
brane.
5.1.1.1 A proposal for radion stabilization In the previous section, we have already mentioned that the internal metric component
G44 gives rise to a massless ﬁeld in an eﬀective description. This means that its vev
rc is very sensitive against any perturbation and rather unstable. For the discussion
of the hierarchy problem it is important that the distance of the branes rc is of the
order of the Planck length. Therefore, it is desirable to stabilize this distance, i.e.
to give a mass to G44 in the eﬀective description. In the present section we brieﬂy
present a proposal of Goldberger and Wise how a stabilization might be achieved via
an additional scalar living in the bulk. We will neglect the back reaction of the scalar
ﬁeld on the geometry. This means that we just consider a scalar ﬁeld in the RS1
background constructed in the previous section. The action consists out of three parts
S = Sbulk + Sb1 + Sb2 , (5.1.1.21) 5. The RS1 model with two branes 172 where Sbulk deﬁnes the ﬁve dimensional dynamics of the ﬁeld and Sb1 and Sb2 its
coupling to the respective branes. We choose
Sbulk = 1
2 d4 x π
−π √
dφ −G GM N ∂M Φ∂N Φ − m2 Φ2 , (5.1.1.22) where Φ is the scalar ﬁeld and GM N is given in (5.1.1.14). The coupling to the branes
is taken to be
Sb1 = − 2
d4x −g b1 λ1 Φ2 − v1 2 Sb2 = − 2
d4x −g b2 λ2 Φ2 − v2 2 , (5.1.1.23) , (5.1.1.24) where vi and λi are dimensionfull parameters whose values will be discussed below.
With the ansatz that Φ does not depend on the xµ for µ = 0, . . . , 3 the equation of
motion for the scalar is
e−4krc φ − e4krc φ
∂φ e−4krc φ ∂φ Φ + m2 Φ
2
rc 2
+4λ1Φ Φ2 − v1 With ν = 4+ m2
k2 δ (φ)
2 δ (φ − π )
+ 4λ2Φ Φ2 − v2
rc
rc = 0. (5.1.1.25) the solution inside the bulk 0 < φ < π is written as
Φ = e2krc φ Aekrc ν φ + Be−krc ν φ , (5.1.1.26) where the integration constants A and B will be ﬁxed below. Plugging this solution
back into the Lagrangian yields an rc dependent constant, i.e. a potential for the
distance of the two branes,
V (rc) = k (ν + 2) A2 e2νkrc π − 1 + k (ν − 2) B 2 1 − e−2νkrc π
2
+λ1 Φ (0)2 − v1 2 2
+ λ2e−4krc π Φ (π )2 − v2 2 . (5.1.1.27) Because of the dependence of Φ on the modulus function (see ﬁgure 5.1) the second
derivative in the ﬁrst term in (5.1.1.25) will lead to delta functions whose argument
vanishes at the position of the branes. Matching this with the delta function source
terms in (5.1.1.25) yields equations for the integration constants A and B . Instead
of writing down and solving those equations explicitly we suppose that λ1 and λ2 are
large enough for the approximation
Φ (0) = v1 , Φ (π ) = v2 (5.1.1.28) 5. The RS1 model with two branes 173 to be suﬃciently accurate. In this approximation one obtains
A = v2 e−(2+ν )krc π − v1 e−2νkrc π , (5.1.1.29) B = v1 1 + e−2νkrc π − v2 e−(2+ν )krc π . (5.1.1.30) The next approximation lies in the assumption that
= m2
4k 1. (5.1.1.31) In evaluating the potential V (rc ) (5.1.1.27), we neglect terms of order
treat krc as a small number. This yields
2
V (rc ) = k v1 + 4ke−4krc π v2 − v1e− −k v1 e −(4+ )krc π krc π 2 − krc π 2v2 − v1 e 1+
. 2 but do not 4
(5.1.1.32) Up to orders of , this potential has a minimum at
krc = 4k2
log
πm2 v1
v2 . (5.1.1.33) In ﬁgure 5.2, we have drawn the potential in a neighborhood of the minimum (using
2
Maple). (What is actually drawn is V − k v1 .) With the appropriate choice for the
scales, the minimum of the potential is clearly visible. One should note, however, the
exponentially suppressed height of the right wall of the potential. If we had chosen
a larger scale for the drawing, ﬁgure 5.2 would just show a runaway potential which
rapidly reaches its asymptotic value. This might be a drawback of the stabilization
mechanism.
The expression for the stable distance between the branes (5.1.1.33) shows that
no extreme ﬁne tuning is needed in order to obtain the wanted value of about ten
for krc . It remains to investigate whether the various approximations (including the
neglection of the back reaction) are sensible. This investigation has been carried out
in[215] by estimating the size of next to leading order corrections. The result is that
the approximations are ﬁne.
To close this section, we should mention that the described stabilization method
is often called “Goldberger Wise mechanism” in the literature. We preferred to use
the term “proposal” because we are not certain that this mechanism is the commonly
established method for solving the problem of moduli stabilization. We decided to
present a brief description of the method because it is one of the most prominent lines
of thought in the context of the Randall Sundrum model. In general, the problem of
moduli stabilization is not very well understood. 5. The RS2 model with one brane 174 6e–09 4e–09 2e–09 0 0.15 0.16 0.17
x 0.18 0.19 0.2 –2e–09 –4e–09 –6e–09 Figure 5.2: The Goldberger Wise potential for k = 10, m = 9, v2 = 1, v1 = 3. The
2
vertical axis shows V − k v1 whereas on the horizontal axis rc is drawn. 5.1.2 The RS2 model with one brane In this section we are going to consider a variant of the model presented in section
5.1.1 where the second brane is removed. Since for the solution of the hierarchy it
was essential that the observers live on this second brane, we now give up the goal of
solving the hierarchy problem (at least temporarily). The construction of the single
brane solution is very simple. The extra dimension is not compact anymore and
therefore we use the coordinate y instead of φ. We do not impose the periodicity
condition (5.1.1.5) but still require a Z2 symmetry under
y → −y. (5.1.2.34) Further, we remove Sb2 from the action (5.1.1.1). Since the extra dimension is not
compact, we can perform rescalings of y in order to remove the rc dependence of the
ansatz (5.1.1.8). Without loss of generality we take rc = 1. Thus, in the single brane
case, the solution for the metric is
ds2 = e−2ky ηµν dxµ dxν + dy 2. (5.1.2.35) 5. The RS2 model with one brane 175 With a non compact extra dimension, one may worry that gravity is ﬁve dimensional
now. However, taking the rc → ∞ limit of (5.1.1.17), one ﬁnds that the eﬀective
four dimensional Planck mass is ﬁnite. This means that the graviton zero mode is
normalizable and yields a four dimensional Newton law. Apart from the zero modes,
there will be also massive gravitons who lead to corrections of Newton’s law. In the
following subsection we will investigate these corrections.
5.1.2.1 Corrections to Newton’s law The Newton potential is obtained by studying ﬂuctuations around the background
(5.1.2.35), for example
G00 = −e−2ky − V (x, y ) , (5.1.2.36) where V denotes a ﬂuctuation. In the presence of a point particle with mass µ at the
origin, the non relativistic limit of the linearized equation for V reads
2
∆3 + e−2ky ∂y + 4kδ (y ) − 4k2 V (x, y ) = Gµ δ (3) (x) δ (y ) , (5.1.2.37) where G is the ﬁve dimensional Newton constant. The fact that V is indeed the Newton
potential can be conﬁrmed by studying the geodesic equation of a point particle probe
and comparing it with the Newton equation of motion. The equation (5.1.2.37) is the
warped geometry analogon of equation (4.4.1.16). (The normalization of the higher
dimensional Newton constant is not really important here.) It is useful to redeﬁne the
coordinate y according to
z≡ sgn (y ) ky
e
−1 .
k (5.1.2.38) With
¯
V = V (x, y ) e k y 
2 (5.1.2.39) equation (5.1.2.37) takes the form
2
∆3 + ∂z − 15k2
¯
+ 3kδ (z ) V = Gµ δ (3) (x) δ (z )
4 (k z  + 1)2 (5.1.2.40) Analogous to section 4.4.1 we plan to expand the solution V into a series of eigenfunctions, i.e. in the case at hand we are looking for solutions of the diﬀerential equation
2
∂z − 15k2
+ 3kδ (z ) ψ (m, z ) = −m2 ψ (m, z ) ,
2
4 (k z  + 1) (5.1.2.41) 5. The RS2 model with one brane 176 where we expect a continuous eigenvalue m now, since the “internal space” is not
compact. Let us discuss ﬁrst the zero mode, i.e. the solution to (5.1.2.41) with m2 = 0.
The zero mode is found to be2
ψ0 (z ) ≡ ψ (0, z ) = N0
3 (k z  + 1) 2 , (5.1.2.42) where N0 is an integration constant to be ﬁxed later. Note that
∂z z  = sgn (z ) , ∂z sgn (z ) = 2δ (z ) . (5.1.2.43) Now, we take m > 0. For z > 0 the general solution to the above equation can be
written as a superposition of Bessel functions
ψ (m, z ) = z  + 1
k c1J2 m z  + 1
k + c2 Y2 m z  + 1
k , (5.1.2.44) where Jν denotes the Bessel functions of the ﬁrst kind whereas Yν stands for the
Bessel functions of the second kind and c1,2 are constants to be ﬁxed below. Because
the solution (5.1.2.44) is written as a function of z , the second derivative with respect
to z in (5.1.2.41) will yield a term containing a δ (z ) (and other terms). One can ﬁx
the ratio c1/c2 by matching the factor in front of this delta function with the factor in
front of the delta function in (5.1.2.44). We will do this in an approximate way. The
most severe corrections to Newton’s law are to be expected from gravitons with small
m (because they carry interactions over longer distances). In matching the coeﬃcients
of the delta functions, only a neighborhood around z = 0 matters. Therefore, we
replace the Bessel functions by their asymptotics for small arguments, which are
J2
Y2 1
m z  +
k
1
m z  +
k 2 1
m2 z  + k
∼
,
8
4
∼−
1
πm2 z  + k (5.1.2.45)
2 − 1
.
π (5.1.2.46) Plugging the asymptotic approximation into (5.1.2.44) and then into (5.1.2.41) one
ﬁnds that the overall coeﬃcient in front of the delta function vanishes if
4k2
c1
=
.
c2
πm2 (5.1.2.47) Hence, our general solution (5.1.2.44) reads
ψ (m, z ) = Nm z  + 1
1
Y2 m z  +
k
k + 1
4k2
J2 m z  +
πm2
k , (5.1.2.48) 2
In forthcoming expressions we will always imply that m > 0 when writing ψ (m, z ). The zero
mode will be denoted by ψ0 (z ) from now on. 5. The RS2 model with one brane 177 where we replaced c2 = Nm because this remaining integration constant will turn out
to depend on the eigenvalue m.
Recall that the extra dimension y (or z ) is not compact. Thus the eigenvalue m is
continuous. Therefore, we normalize
dz ψ (m, z ) ψ m , z = δ m − m , (5.1.2.49) for m, m > 0. For m ≥ 0 we impose the normalization condition
dz ψ0 (z ) ψ (m, z ) = δm,0 , (5.1.2.50) such that the completeness relation reads
∞ ψ0 (z ) ψ0 z + dm ψ (m, z ) ψ m, z 0 =δ z−z . (5.1.2.51) The orthonormalization condition (5.1.2.49) ﬁxes Nm . It turns out that the computation simpliﬁes essentially in the approximation where the arguments of the Bessel
functions are large, since the corresponding asymptotics yields plane waves. Explicitly,
for large mz the Bessel functions are approximated by
√
zJ2 (mz ) ∼ 2
5π
cos mz −
πm
4 , (5.1.2.52) √ 2
5π
sin mz −
πm
4 . (5.1.2.53) zY2 (mz ) ∼ Because we are mainly concerned about large (> µm
1/Mp ) distance modiﬁcations
2
of Newton’s law we focus on the contribution of the “light” modes ( m2
1). (Recall
k
that k is of the order of the Planck mass.) Then (5.1.2.49) yields for the normalization
constant (for m > 0)
5 Nm πm 2
=
.
(4k2) (5.1.2.54) The condition (5.1.2.50) is satisﬁed for m > 0 to a good approximation. Evaluating
(5.1.2.50) for m = 0 ﬁxes
N0 = √
k. (5.1.2.55) ¯
Now, we expand V (x, z ) into eigenfunctions ψ0 (z ) and ψ (m, z ) with x dependent
coeﬃcients ϕm (x)
∞ ¯
V (x, z ) = ϕ0 (x) ψ0 (z ) +
0 dm ϕm (x) ψ (m, z ) . (5.1.2.56) 5. The RS2 model with one brane 178 By plugging the ansatz (5.1.2.56) into (5.1.2.40), we ﬁnd that for m ≥ 0 and r = x
ϕm (x) = − Gµ −mr
e
am ,
r (5.1.2.57) with the constants am taken such that
a0 ψ0 (z ) + dm amψ (m, z ) = δ (z ) . (5.1.2.58) Comparison with (5.1.2.51) yields
a0 = ψ0 (0) , am = ψ (m, 0) . (5.1.2.59) In the current setup we are interested in corrections to Newton’s law as an observer on
the brane at the origin would measure them. Deﬁning the four dimensional Newton
constant G4 as
G4 = Gk, (5.1.2.60) we ﬁnd from (5.1.2.56)
G4 µ
¯
V (x, 0) = V (x, 0) = −
r ∞ 1+ dm 0 m −mr
e
,
k2 (5.1.2.61) where once again we took into account only modes with m/k
1 such that we could
use the asymptotics (5.1.2.45) and (5.1.2.46) in order to evaluate ψ (m, 0). Finally,
performing the integral in (5.1.2.61) leads to
V (x, 0) = −G4 µ
r 1+ 1
r2k2 . (5.1.2.62) For k being of the order of the Planck mass (5.1.2.62) is in very good agreement
with the experimental values. This may look a bit surprising. Even though the extra
dimension is not compact, we obtain a four dimensional Newton potential for observers
who live on the brane at y = 0. This non trivial result ﬁnds its explanation in the
exponentially warped geometry. It is this geometry which is responsible for the fact
that the amplitude of the zero mode has its maximum at the brane and vanishes rapidly
for ﬁnite z . On the other hand, the massive modes reach their maximal amplitudes
asymptotically far away from the brane. Therefore, they have very little inﬂuence on
the gravitational interactions on the brane, although the masses of the extra gravitons
can be arbitrarily small.
In the following subsection we are going to rederive (5.1.2.62) in a diﬀerent way. 5. The RS2 model with one brane
5.1.2.2 179 ... and the holographic principle In section 4.3 we have described a duality between a ﬁeld theory living on the boundary
of an AdS5 space and a theory living in the bulk of an AdS5 space. This correspondence
is sometimes called the holographic principle since it allows to reproduce bulk data
from boundary data (and vice versa). Now we are going to apply this principle to the
RS2 setup. Before doing so, we will establish that the RS2 setup has something to do
with an AdS5 space (namely it is a slice of an AdS5 space). To this end, we ﬁrst write
down the RS2 metric (5.1.2.35) in terms of the coordinate z deﬁned in (5.1.2.38). This
results in
ds2 2 =
RS 1
µ
ν
2
2 ηµν dx dx + dz .
(k z  + 1) (5.1.2.63) For symmetry reasons the coordinate z can be restricted to the half interval between
zero and inﬁnity. The singularity at z = 0 is caused by the brane.
Now, let us recall from section 4.3 that the AdS5 metric is (see (4.3.1.12))
ds2 =
AdS dU 2
U2
ηµν dxµ dxν + R2 2 .
R2
U (5.1.2.64) Changing the coordinates according to (−R < z < ∞)
U= R2
z+R (5.1.2.65) yields an AdS metric of the form
ds2 =
Ads 1
1+ ηµν dxµ dxν
z2
R + dz 2. (5.1.2.66) Comparing (5.1.2.66) with (5.1.2.63), we observe that the RS2 geometry describes a
slice of an AdS5 space. The radius of the AdS5 space is 1/k, and the space is cut oﬀ at
z = 0. Since the boundary of the AdS5 space is situated at U → ∞, the cutoﬀ at z = 0
means that we lost the region between U = R and the boundary. Hence, the position of
the brane in the RS2 setup can be viewed as an infrared cutoﬀ for gravity on an AdS5
space. This suggests that we may apply the AdS/CFT conjecture on the RS2 scenario.
(Note however, that we do not have any supersymmetry now. Without supersymmetry
the AdS/CFT conjecture has passed less consistency checks. Nevertheless, let us
assume that the conjecture is correct also without supersymmetry.) The ﬁeld theory
dual of the RS2 setup is thus a conformal ﬁeld theory with a UV cutoﬀ3 given by
k. (The cutoﬀ actually breaks the conformal invariance. The conformal anomaly
3
Recall from section 4.3.2.1 that an IR cutoﬀ in the bulk theory corresponds to a UV cutoﬀ in the
dual ﬁeld theory. 5. The RS2 model with one brane 180 induces a coupling of the ﬁeld theory to gravity.) In particular, we plan to employ the
AdS/CFT duality conjecture for the computation of corrections to Newton’s law. As
a preparation let us sketch how Newton’s potential is related to the gravity propagator
in four dimensions. If we did not have an extra dimension, the gravity propagator in
2
momentum space is (up to a polarization tensor) 1/ Mp p2 . The Newton potential
can be obtained from this propagator by formally setting the p0 component to zero
and Fourier transforming4 with respect to the spatial momentum components. The
2
result in position space is then 1/(Mp r). Therefore, we will use the AdS/CFT duality
conjecture to compute the corrected graviton propagator and deduce the corrected
Newton potential via the above description.
The dual picture for the RS2 setup is that we have four dimensional gravity plus
the CFT dual of gravity on AdS5 with a UV cutoﬀ k. Corrections to four dimensional
gravity are caused by the interaction of gravity with the CFT. The eﬀective graviton
propagator is obtained by integrating over the CFT degrees of freedom. The one loop
corrected graviton propagator will be schematically of the form
1
2
Mp p2 1 + TCF T (p) TCF T (−p) 1
2
Mp p2 , (5.1.2.67) where TCF T stands for the energy momentum tensor of the CFT dual. (The coupling
of gravity to the CFT ﬁelds is given by the energy momentum tensor.) For any four
dimensional CFT, the two point function of the energy momentum tensor is ﬁxed to
be of the form
TCF T (p) TCF T (−p) = cp4, (5.1.2.68) where we imposed that the UV cutoﬀ is k. We will not derive this result here, but
just give two comments. First, notice that (5.1.2.68) is the four dimensional analogon
of (2.1.3.51). The number c quantiﬁes the conformal anomaly. The second remark
is, that the reader may get some impression on how the expression (5.1.2.68) arises
by computing it explicitly for pure gauge theory. We are interested in the order of
magnitude of c. This has been computed in[241] to be
c≈ 2
3
Mp
M5
= 2,
k3
k (5.1.2.69) where M5 denotes the ﬁve dimensional Planck mass and the radius of the AdS space
dual to the CFT is 1/k. In the second equality of (5.1.2.69) we used the relation
4 We use the following prescription for performing the Fourier transformation. Transforming the
equation ∆3 f (x) = δ (3) (x) one ﬁnds that 1/p2 transforms into 1/r. Later we will also have to compute
Fourier transforms with additional powers of p in the numerator or denominator. An additional power
of p in the numerator transforms into ∂r whereas powers of p in the denominator can be generated by
∂p which in turn transforms into r. 5. The RS2 model with one brane 181 between the ﬁve and four dimensional Planck mass (M5 and Mp ) which, in the RS2
setup, is obtained by taking rc → ∞ in (5.1.1.17). The corrected Newton potential is obtained by setting formally p0 to zero in
(5.1.2.67) and performing a three dimensional Fourier transformation to the position space. Thus, the eﬀect of integrating over the CFT ﬁelds results in the following
replacement of the Coulomb (Newton) potential
− 1
1
→−
r
r 1+ 1
k2 r2 . (5.1.2.70) This result agrees with the expression (5.1.2.62) computed in the previous section.
Thus, we have learned that integrating over the CFT ﬁelds yields the same corrections
to four dimensional gravity as taking into account the massive “KaluzaKlein” gravitons. Employing the AdS/CFT correspondence, the computational eﬀort decreases
substantially. We will make use of this fact when we combine the RS1 with the RS2
scenario in the next subsection.
5.1.2.3 The RS2 model with two branes In the previous two subsections we have seen that the RS2 setup has the exciting
feature of giving rise to eﬀectively four dimensional gravitational interactions even
though the extra dimension is not compact. On the other hand, we observed before
that the RS1 model is capable to explain the hierarchy between the Planck scale and
the weak scale without introducing large numbers. How can we combine these two
models? We should introduce a brane with the observers at y = πrc into the RS2
setup. However, this brane should not cause a change of the RS2 metric (5.1.2.35).
The observers on the additional brane (at y = πrc ) can achieve this by performing a
ﬁne tuning such that the vacuum energy on their brane vanishes. In the following we
will call the brane at y = πrc the SM (Standard Model) brane. The SM brane can
be viewed as a probe in the RS2 background. The hierarchy can now be explained in
the same way as it is explained in the RS1 setup. What we should worry about are
the gravitational interactions as viewed by an observer on the SM brane. In principle,
these can be computed along the lines of section 5.1.2.1. The situation is, however,
slightly more complicated since the approximation has to be reﬁned. In particular,
replacing the Bessel functions by their plane wave asymptotics in the computation
of Nm is too rough an estimate. Now, this would imply that the observer on the
SM brane sees the Bessel functions as plane waves. As argued in[332] this is not the
case, in particular for the light continuum modes. The authors of[332] reﬁned the 5. The RS2 model with one brane 182 approximation and obtained the result
V (r, y = πrc ) = − G4 µ
r 1+ 1
k2 r2 − µ
8
Mw r7 (5.1.2.71) for the Newton potential observed on the SM brane. Here, Mw is of the order of a TeV
if we take rc such that the hierarchy problem is solved. Instead of going through the
tedious reﬁnement of the approximations performed in section 5.1.2.1, we employ the
AdS/CFT correspondence to motivate (5.1.2.71). The introduction of the SM brane
modiﬁes the RS2 dual such that it consists out of four dimensional gravity, the CFT
dual of the RS2 AdS5 slice and the Standard Model of the probe brane. Note that
yc = rc π is translated to U0 − Uc = T eV in the course of the coordinate transformations
(5.1.2.38) and (5.1.2.65), where U0 denotes the position of the brane at the origin and
Uc the position of the SM brane. This means that SM ﬁelds and CFT ﬁelds interact via
ﬁelds with masses of the order of a T eV .5 Integrating out those ﬁelds yields eﬀective
coupling terms between SM ﬁelds and CFT ﬁelds. (This is analogous to generating
the Fermi interaction via integrating out the W and Z bosons.) The structure of the
possible interaction terms is restricted by symmetries to[37]
1 µν
T TµνCF T .
4
Mw SM (5.1.2.72) Note the similarity to the coupling of the SM ﬁelds to gravitons. Apart from charged
interactions, the SM ﬁelds interact via gravitons and via CFT ﬁelds. This suggests
that for an observer on the SM brane the eﬀective graviton propagator is
1
2
Mp p2 1 + TCF T (p) TCF T (−p) 1
2
Mp p2 + 1
TCF T (p) TCF T (−p) ,
8
Mw (5.1.2.73) where the ﬁrst two terms are the same as in (5.1.2.67), and the last term means
that the observer will interpret the interaction (5.1.2.72) as gravitational interaction.
In computing the contribution due to the last term we use (5.1.2.68). Applying the
recipe of the previous section, we obtain out of the propagator the modiﬁed Newton
potential (5.1.2.71). This potential is still in agreement with the observational bounds
on deviations from Newton’s law. Hence, adding a probe brane at y = πrc in the
RS2 setup one obtains a model which explains the hierarchy and possesses eﬀectively
four dimensional gravitational interactions, even though there is a non compact extra
dimension. However, we should remark that we discussed the setup only classically and
showing its stability against quantum corrections may be a problematic issue. This
5
One may view the ﬁeld theory dual as a stack of Dbranes on which the CFT lives and the SM
probe brane separated by a distance 1/T eV from the CFT branes. The interaction between the CFT
and the SM can be thought of as arising due to open strings stretching between the corresponding
branes. 5. Inclusion of a bulk scalar 183 corresponds to the technical hierarchy problem which can be solved by supersymmetry
in conventional four dimensional models. Supersymmetric versions of the RS model
appear in the literature listed in chapter 6. 5.2 Inclusion of a bulk scalar In this section, we are going to modify the Randall Sundrum models of the previous
section by introducing a bulk scalar Φ which couples also to the branes. Actually, we
have considered this modiﬁcation already in section 5.1.1.1, where we neglected the
back reaction of the scalar on the geometry. In the current section we are going to
take this back reaction into account. We will not return to the stabilization mechanism
of section 5.1.1.1, though. (The inclusion of back reaction into the Golberger Wise
mechanism is discussed in[128], with the result that the mechanism works also when
the back reaction is included.) Instead of addressing the question of how a scalar
helps to stabilize the inter brane distance, we want to consider another question. As
we will see the cosmological constant problem is reformulated in a brane world setup.
We will investigate whether a scalar can help to ﬁnd a solution to the cosmological
constant problem. Before doing so, we brieﬂy present a solution generating technique
and consistency conditions on the solutions. 5.2.1 A solution generating technique Introducing a bulk scalar Φ modiﬁes the action (5.1.1.1) to6
S= d4x √
4
dy −G R − (∂ Φ)2 − V (Φ) −
3 i bi d4 x −g bi fi (Φ) , (5.2.1.1) where y is the coordinate labeling the extra dimension, and the sum over i stands for a
sum over the branes. The index bi at the integral means that y is ﬁxed to the position
(yi ) of the brane bi . The function V (Φ) is a bulk potential for the scalar and fi (Φ) is
the coupling function of the scalar to the brane bi .
For later use let us also generalize the ansatz (5.1.1.8) to
ds2 = e2A(y) gµν dxµ dxν + dy 2,
¯ (5.2.1.2) where gµν denotes the metric of a four dimensional maximally symmetric space, i.e.
¯ for M4 diag (−1, 1, 1, 1) √
√
√ ¯ t 2 Λ t 2 Λt
¯
¯
diag −1, e2 Λ , e
,e
for dS4
gµν =
¯
.
(5.2.1.3)
√
√
√ ¯ x 3 2 −Λ x 3 2 −Λ x 3
¯
¯ diag −e2 −Λ , e
,e
, 1 for AdS
4 6
For simplicity we set the ﬁve dimensional Planck mass to one. It can be introduced if needed by
a simple analysis of the mass dimensions. 5. A solution generating technique 184 ¯
The constant Λ is related to the constant curvature of the de Sitter (dS4) and the anti
de Sitter (AdS4) slices.
¯
Let us ﬁrst discuss the simplest case with Λ = 0. As usual we consider ﬁelds which
depend only on y and denote a derivative with respect to y by a prime. The equations
¯
of motion for Λ = 0 are
8
32
∂V
Φ + AΦ −
−
3
3
∂Φ
6A
3A + 4
Φ
3 2 + ∂fi
δ (y − yi ) = 0,
∂Φ i
2 1
2 (5.2.1.4) 2
Φ
3 = 0, (5.2.1.5) fi δ (y − yi ) = 0. (5.2.1.6) −
i 2 + V
2 First, we analyze this system of equations in absence of the branes. We start with the
ansatz
A = W (Φ) . (5.2.1.7) Equation (5.2.1.6) ﬁxes then
Φ =− 9 ∂W
.
4 ∂Φ (5.2.1.8) the second equation (5.2.1.5) yields
27
V=
4 ∂W
∂Φ 2 − 12W 2. (5.2.1.9) Finally, the ﬁrst equation (5.2.1.4) is satisﬁed automatically.
With view on (5.2.1.9), we could formally call W a superpotential because such a
relation is known from ﬁve dimensional gauged supergravity[192]. A solution in the
absence of branes can now be constructed as follows. Equation (5.2.1.9) determines
W up to an integration constant. With a given W , one can solve (5.2.1.8) for Φ up to
another integration constant. Equation (5.2.1.7) ﬁxes A up to an integration constant.
So altogether, there are three integration constants in the general solution.
Now, we take into account the source terms caused by the presence of the branes.
We are looking for solutions in which the ﬁelds are continuous. Therefore, the ﬁrst
derivatives of the ﬁelds A and Φ are ﬁnite arbitrarily close to the position of the branes.
However, the ﬁrst derivatives must jump when y passes a yi . Integrating (5.2.1.6) and
(5.2.1.4) over y = yi − . . . yi + and taking the limit → 0, one ﬁnds the jump
conditions
3 A (yi + 0) − A (yi − 0)
8
Φ (yi + 0) − Φ (yi − 0)
3 1
= − fi ,
2
∂fi
=
.
∂Φ (5.2.1.10)
(5.2.1.11) 5. A solution generating technique 185 For the “superpotential” W , this implies
3 Wy=yi +0 − Wy=yi −0
3
2 ∂W
∂W
−
∂ Φ y=yi +0
∂ Φ y=yi −0 1
= − fi ,
2
∂fi
.
=−
∂Φ (5.2.1.12)
(5.2.1.13) This means that there are two additional conditions per brane. If we safely want to
obtain four dimensional gravity in the eﬀective theory, we should compactify the extra
dimension. For an interval compactiﬁcation we need at least two branes. The length
of the interval (the inter brane distance) enters the ansatz as a further integration
constant (e.g. rc in (5.1.1.8) now appears in (5.1.1.5)). Therefore, four integration
constants are to be ﬁxed by four conditions. However, we should take into account
that one of the integration constants corresponds to constant shifts in A which can
be absorbed into a rescaling of x. A enters the equation of motions and the jump
conditions only with its derivatives. Therefore, one of the integration constants is not
ﬁxed by the jump conditions. This means that in a two (or more) brane setup at least
one ﬁne tuning of the model parameters (appearing in V (Φ) and fi (Φ)) is necessary
¯
for the existence of a solution with Λ = 0.
For example in the RS1 model, we obtained two ﬁne tuning conditions (5.1.1.13).
The fact that there is one more ﬁne tuning condition than expected by naive counting
is related to the fact that the inter brane distance rc is a modulus of the solution. This
feature is closely connected with the observation that we can remove the second brane
and still obtain four dimensional eﬀective gravity. Even after removing one brane the
Randall Sundrum model requires one ﬁne tuning. We will come back to this point in
section 5.2.3.
The fact that our solution requires ﬁne tuning of parameters has its origin in the
¯
¯
Λ = 0 condition of the ansatz we have considered so far. We can view Λ as an
additional integration constant in the ansatz (5.2.1.2). In general, constant shifts in
¯
A can be absorbed in a rescaling of xµ in combination with a redeﬁnition of Λ. This
¯
suggests that a mismatch in the ﬁne tuning conditions results in a nonzero Λ. In order
¯
to see this more explicitly we write down the equations of motion for Λ = 0,
32
∂V
8
Φ + AΦ −
−
3
3
∂Φ ∂fi
δ (y − yi ) = 0,
∂Φ i 2
V
2
¯
6A −
Φ + − 6Λe−2A = 0,
3
2
1
4
2
¯
Φ + 3Λe−2A +
fi δ (y − yi ) = 0.
3A +
3
2
2 (5.2.1.14)
(5.2.1.15)
(5.2.1.16) i The jump conditions (5.2.1.10) and (5.2.1.11) are still of the same form. We observe
that a constant shift in A enters the equations of motion. Hence, there is no ﬁne 5. Consistency conditions 186 ¯
tuning to be expected if we do not ﬁx the value of Λ in the ansatz. For completeness,
we note that the equations of motion can be reduced to a set of ﬁrst order equations
¯
like in the Λ = 0 case. The corresponding ﬁrst order equations are
27 1
4 γ (r)2 ∂ W (Φ)
∂Φ V = A 2 − 12W (Φ)2 , = γ (r) W (Φ) ,
9 1 ∂W (Φ)
,
=−
4 γ (r) ∂ Φ Φ γ (r) = 1+ ¯
Λ
e−2A .
W (Φ)2 (5.2.1.17)
(5.2.1.18)
(5.2.1.19)
(5.2.1.20) To ﬁnd a solution to this system of ﬁrst order equations looks more complicated than
¯
in the Λ = 0 case. The equation (5.2.1.17) now couples to the rest of the equations
due to the γ dependent factor. 5.2.2 Consistency conditions In this subsection we are going to discuss consistency conditions which any solution
to the setup of the previous subsection has to satisfy. In principle, these consistency
conditions constitute nothing but a check whether there has been a computational
error. They are, however, useful in cases where the envisaged solution possesses singularities. Further, consistency conditions give sometimes informations about the system
without the need of constructing an explicit solution. The condition we are going do
derive next is most simply expressed in words. It states that the four dimensional
eﬀective cosmological constant is compatible with the constant curvature of the four
¯
dimensional slices. (This curvature is ﬁxed by Λ in (5.2.1.3).) Now let us translate
this verbal statement into fromulæ.
In order to obtain the four dimensional eﬀective cosmological constant, we need
to construct an eﬀective action for four dimensional gravity. We start with the ﬁve
dimensional metric
ds2 = e2A(y) gµν dxµ dxν + dy 2,
˜ (5.2.2.1) gµν = gµν + hµν
˜
¯ (5.2.2.2) where is the metric on the four dimensional slices. It is taken to be independent of y , and the
background metric gµν is deﬁned in (5.2.1.3). If we do not consider other ﬂuctuations
¯
than hµν , the action for four dimensional gravity will be of the general form
2
S4 = Mp ˜˜
d4x −g R(4) − λ , (5.2.2.3) 5. Consistency conditions 187 ˜
where R(4) is the four dimensional scalar curvature computed from gµν . The cosmo˜
logical constant λ is ﬁxed by the condition that gµν = gµν should be a stationary point
˜
¯
of (5.2.2.3). This yields
¯
λ = 6Λ. (5.2.2.4) We should also recall that the eﬀective four dimensional Planck mass is given by
2
Mp = dy e2A(y) , (5.2.2.5) where A takes its classical value. The vacuum value of the Lagrange density in (5.2.2.3)
can be easily computed to be
2¯
¯2
¯
L4 = Mp R(4) − λ = 6ΛMp , (5.2.2.6) ¯
where R(4) is the scalar curvature computed from gµν .
¯
¯4 should coincide with a result obtained in the following way.
For consistency, L We plug the solution of the equations of motion into the ﬁve dimensional action and
integrate over y . (This is exactly the prescription of obtaining the classical value of
the four dimensional Lagrangian.) In order to do so, it is useful to write down part of
the equations of motion in a less explicit form than before. The equations obtained
from ﬁve dimensional metric variations are the ﬁve dimensional Einstein equations
1
1
RM N − GM N R = TM N .
2
2 (5.2.2.7) For the model deﬁned in (5.2.1.1), the energy momentum tensor TM N is
TM N = 8
4
∂M Φ∂N Φ − (∂ Φ)2 GM N − V (Φ) GM N −
3
3 i µν
fi δ (y − yi ) gµν δM δN , (5.2.2.8)
where gµν is the metric induced on the brane (see (5.1.1.3)). The classical value of R
can be easily computed by taking the trace of (5.2.2.7) with the result
R= 4
5
4
(∂ Φ)2 + V (Φ) +
3
3
3 i fi δ (y − yi ) . (5.2.2.9) Plugging this into (5.2.1.1), we obtain the classical value for the four dimensional
Lagrangian
¯
L4 = dy e4A 2
1
V (Φ) +
3
3 i fi δ (y − yi ) , (5.2.2.10) 5. The cosmological constant problem 188 where it is understood that A and Φ satisfy the equations of motion. Comparing with
(5.2.2.6) and using (5.2.2.8), we obtain ﬁnally the consistency condition
− 1
3 0
5
¯2
dy e4A T0 + T5 = 6ΛMp . (5.2.2.11) We should emphasize again that (5.2.2.11) is just a consequence of the equations of
¯
motion. For Λ = 0, (5.2.2.11) implies that the vacuum energy density of the solution
has to vanish.
Before closing this subsection we want to describe an alternative way to obtain the
same (or equivalent) consistency conditions. First, we note that
A enA = enA n−4
Φ
9 2 − 1
nV
¯
+ (n − 1) Λe−2A−
12
6 i fi δ (y − yi ) . (5.2.2.12) This can be easily checked with the equations (5.2.1.15) and (5.2.1.16). With the
expression (5.2.2.8) we rewrite (5.2.2.12) in the following way
A enA = enA 10
T+
60 n
1
−
12 6 5
¯
T5 + (n − 1) Λe−2A . (5.2.2.13) Assuming that for a consistent solution the integral over the total derivative on the
lhs of (5.2.2.13) vanishes we ﬁnd
− 1
3 0
dy enA T0 + n
5
¯
− 1 T5 = 2 (n − 1) Λ
2 dy e(n−2)A . (5.2.2.14) We observe that for n = 4 this condition is identical to the previously derived condition
(5.2.2.11). In the next subsection we will discuss solutions with singularities. For those
solutions, one could argue that the preposition of condition (5.2.2.14) is not necessarily
satisﬁed. If there are singularities, an integral over a total derivative may diﬀer from
zero, and one may not worry about (5.2.2.14) in such a case. For n = 4, we have shown
that (5.2.2.14) encodes the statement that the eﬀective four dimensional cosmological
constant is compatible with the curvature of the four dimensional slices. This should
be the case also in the presence of singularities. We leave it as an exercise to verify
that the Randall Sundrum models satisfy all the consistency conditions. 5.2.3 The cosmological constant problem In this section, we are going to discuss whether it is possible to solve the cosmological
constant problem within a brane world scenario containing a bulk scalar. Let us ﬁrst
state the problem as it arises in conventional quantum ﬁeld theory. The observational
bound on the value of the cosmological constant (as measured from the curvature of
the universe) is
2
λMp ≤ 10−120 (Mp )4 . (5.2.3.1) 5. The cosmological constant problem 189 Taking into account the leading order contribution of quantum ﬁeld theory, one obtains
2
2
λMp = λ0Mp + (UVcutoﬀ)4 Str (1) , (5.2.3.2) where λ0 corresponds to a tree level contribution which can be viewed as an input
parameter of the model. The size of the UVcutoﬀ is set by the scale up to which
the eﬀective ﬁeld theory at hand is valid. The supertrace is taken over degrees of
freedom which are light compared to the UVcutoﬀ. If for example we assume that
the standard model of particle physics is a valid eﬀective description of physics up
2
to the Planck scale, we need to ﬁne tune 120 digits of the input parameter λ0 Mp in
order to obtain agreement with (5.2.3.1). The situation slightly improves if we assume
that the standard model is a good eﬀective description only up to a supersymmetry
breaking scale at (at least) about a TeV. In this case we should take the UVcutoﬀ
2
to be roughly a TeV. We still have to ﬁne tune 60 digits in λ0Mp in order to match
the observation (5.2.3.1). To summarize, the cosmological constant problem is that a
huge amount of ﬁne tuning of input parameters is implied by the observational bound on the cosmological constant.
How could the situation improve in a brane world setup? Here, it may happen
that the ﬁeld theory produces a huge amount of vacuum energy which however results
only in a curvature along the invisible extra dimension. In section 5.2.1 we have seen
that in a two (or more) brane setup we need to ﬁne tune input parameters such that
¯
¯
Λ = 07 is a solution of the model. (See equation (5.2.2.4) for the relation between Λ
and λ.) Actually, the amount of ﬁne tuning needed in a two brane setup is of the order
of magnitude by which the vacuum energy on a brane deviates from the observed value
(5.2.3.1) because this quantity enters the jump conditions. One may hope to ﬁnd a
single brane model for which a solution without ﬁne tuning exists. This possibility is
not excluded by our investigations in section 5.2.1. However, we will prove later that a
single brane model with eﬀectively four dimensional gravity requires a ﬁne tuning (as
the RS2 model of section 5.1.2 does). Before presenting the general (negative) result,
we would like to demonstrate the problems at an illustrative example.
5.2.3.1 An example The model we are going to discuss is a special case of (5.2.1.1) with a single brane at
y = 0 as well as V (Φ) ≡ 0 and f0 (Φ) = T ebΦ . Hence, the action reads
S= √
4
d5 x −G R − (∂ Φ)2
3 − √
d4x −gT ebΦ y=0 , (5.2.3.3) 7
Recent observations seem to hint at a small but non zero constant. For the discussion of the ﬁne
tuning problem this value is too small to be relevant. 5. The cosmological constant problem 190 4
where b and T are constants. In what follows we will focus on the case b = ± 3 . The
case b = ± 4 is similar and discussed in[268, 267], [36], [186, 187]. We take the ansatz
3
¯
(5.2.1.2) with Λ = 0. From equation (5.2.1.5) one ﬁnds that 1
A =± Φ.
3 (5.2.3.4) We choose
1
3Φ ,
1
−3Φ A= y<0
, y>0 . (5.2.3.5) The reader may verify that taking the same sign on both sides of the brane does not
lead to a consistent solution. The only other choice is to interchange the signs in
(5.2.3.5). This can be undone by redeﬁning y → −y and hence the ansatz (5.2.3.5)
4
is general (for b = ± 3 ). The rest of the equations of motion is easily solved with the
result Φ(y ) = 4
3 y + c 1 + d1 ,
3
− 4 log 4 y + c2 + d2,
3
3
4 log y<0
y>0 , (5.2.3.6) where ci and di are integration constants. The condition that Φ should be continuous
at y = 0 ﬁxes d2 in terms of the other integration constants. The jump conditions
(5.2.1.10) and (5.2.1.11) determine c1 and c2 in terms of d1 according to
2
c2
2
c1 =
= 3b 1
−
8
2
3b 1
−+
8
2
− 3b T ebd1 c1 4 ,
3b T ebd1 c1 4 . (5.2.3.7)
(5.2.3.8) Together with possible constant shifts in A, two integration constants are not ﬁxed by
the equations of motion.
The next step is to ensure that an observer will experience four dimensional gravitational interactions (plus possible small corrections). This is the case only if the four
dimensional Planck mass is ﬁnite. The expression for the four dimensional Planck
mass is given in (5.2.2.5). If the parameters (T and b) of the model are such that
there is no singularity at some y > 0 (y < 0) the integration region in (5.2.2.5) extends
to (minus) inﬁnity. In one or both of these cases the four dimensional Planck mass
diverges, and an eﬀective four dimensional theory decouples from gravity. This is not
what we are interested in since with decoupled gravity the problem of the cosmological
constant does not occur. Therefore, we have to choose our parameters such that there
are singularities at which we can cut oﬀ the integration over y . Explicitly this imposes 5. The cosmological constant problem 191 the conditions
T
T 1 3b
−
2
8
1 3b
−−
2
8 > 0,
< 0. (5.2.3.9) These conditions are easy to satisfy without ﬁne tuning of the parameters. So far,
it looks as if we have achieved to ﬁnd a solution with vanishing four dimensional
curvature without the necessity of a severe ﬁne tuning of input parameters.
It remains to check whether the consistency condition (5.2.2.11) is satisﬁed. Since
¯
we have taken the ansatz with Λ = 0, the condition states that the vacuum energy
density of our solution should vanish. The vacuum energy density is most easily
computed from (5.2.2.10). To be speciﬁc, we ﬁx the integration constant in A via
1
A = 3 Φ for y < 0. Taking further into account that our background is static, we ﬁnd
for the vacuum energy density
4
28
1
e 3 d1 = 0.
E = − T e4A+bΦ y=0 = −
3
3 4 − 3b (5.2.3.10) We see that the consistency condition is not satisﬁed. Since the condition of vanishing
vacuum energy density E = 0 is derived from the equation of motion, (5.2.3.10) implies that the equations of motion are not solved. Indeed, with the parameter choice
(5.2.3.9), the second derivatives of Φ and A contain delta functions which are not canceled by source terms in the equations of motion. We have to cure this inconsistency
by adding additional source terms to the setup, i.e. to extend the single brane scenario
to a three brane scenario. From our considerations in section 5.2.1, we know already
that this will lead to ﬁne tuning conditions on the input parameters. For illustrative
purposes, let us demonstrate the appearance of the ﬁne tuning explicitly. We modify
our action (5.2.3.3) by two additional source terms, i.e.
S → S + S + + S− , (5.2.3.11) with
S± = − d4 xT± eb± Φ y=y± . (5.2.3.12) The quantities b± and T± are now input parameters of the model. The value of y±
gives the locations of the singularities,
3
3
y− = − c1 , y+ = − c2.
4
4 (5.2.3.13) 5. The cosmological constant problem 192 The additional source terms give rise to four more jump conditions to be satisﬁed by
the solution. These jump conditions are
8
Φ (y± + 0) − Φ (y± − 0)
3
Φ (y± + 0) − Φ (y± − 0) = b± T± eb± Φ(y± ) , (5.2.3.14) 1
= − T± eb± Φ(y) .
2 (5.2.3.15) Before solving these additional jump conditions we need to give a prescription how to
continue our solution beyond the singularities. There are several possible descriptions.
For example, one may continue in such a way that the setup becomes periodic in y .
The simplest choice is to eﬀectively cut oﬀ the space at the singularities (at y = y± ) by freezing the ﬁelds to the singularity values for y ∈ [y− , y+ ] such that the ﬁrst derivatives
vanish beyond the singularities. (The ﬁnal conclusion is not aﬀected by the particular
way of continuing the solution beyond the singularities.) With our prescription the
conditions (5.2.3.14) and (5.2.3.15) are solved by
b± = ± 4
3 (5.2.3.16) and
4 4 T− e− 3 d1 = T+ e 3 d2 = −2. (5.2.3.17) One should recall that d2 is already ﬁxed by the jump conditions at y = 0. We observe
that the input parameters need to be ﬁne tuned.
The contribution of the branes at y = y± to the vacuum energy density is
1
T+ e4A+b+ Φ y=y+ + T− e4A+b− Φ y=y−
3
2 4 d1 8
e3
,
3
4 − 3b E+ + E − = −
= (5.2.3.18) where we have employed the jump conditions and ﬁxed an integration constant in
1
A by the choice A = 3 Φ for y < 0. Hence, the contribution (5.2.3.10) is exactly
canceled by the additional branes and the model is consistent now. However, we failed
to construct a brane setup yielding a vanishing eﬀective four dimensional cosmological constant without ﬁne tuning of the parameters. If the ﬁne tuning is not satisﬁed, there
4
¯
exist Λ = 0 solutions[267]. (The situation is slightly diﬀerent in the b = ± 3 case where
¯
the possible value of Λ is ﬁxed by the bulk potential V , which needs to be ﬁne tuned
¯
to zero for Λ = 0 to be a solution[187]. In addition there is a ﬁne tuning due to the
4
necessity of additional branes for b = ± 3 , too.)
¯
In the next subsection we will show that our failure to ﬁnd a Λ = 0 solution without
ﬁne tuning is not caused by an unfortunate choice of the model we started with but rather a generic feature of brane models with a bulk scalar. 5. The cosmological constant problem
5.2.3.2 193 A no go theorem The prepositions for the no go theorem for a “brany” solution to the cosmological
constant problem are:
¯
• The model contains a single brane and Λ = 0.
• The four dimensional Planck mass is ﬁnite.
• The model does not contain singularities apart from the one corresponding to
the single brane source.
• The bulk potential V can be expressed in terms of the “superpotential” W
according to (5.2.1.9).
In a ﬁrst step, we are going to show that these prepositions imply that the ﬁve
dimensional space must be asymptotically (for large y ) an AdS space. Suppose the
warp factor asymptotically shows a power like behavior,
eA ∼ y −α . (5.2.3.19) The four dimensional Planck mass is computed in (5.2.2.5). With a single brane and no
further singularities the integration is taken over y ∈ (−∞, ∞). A necessary condition
for
∞ dy e2A < ∞ (5.2.3.20) 1
α> .
2 −∞ (5.2.3.21) is On the other hand, equation (5.2.1.6) tells us that in the bulk (in particular asymptotically)
A < 0 =⇒ α < 0. (5.2.3.22) We conclude that eA cannot fall oﬀ with a power of y  as y  → ∞.
Therefore, we assume an exponential fall oﬀ, i.e. for large y 
A = −k y  , (5.2.3.23) with k being a positive constant. In the following we will show that in this case there is
a ﬁne tuning similar to the ﬁne tuning of the RS2 model. Before going into the details,
let us sketch the outline of the proof. The asymptotic behavior (5.2.3.23) suﬃces to 5. The cosmological constant problem 194 reproduce the superpotential for all y . Plugging this into the matching conditions
(5.2.1.12) and (5.2.1.13) will show that the input parameters of the model need to be
ﬁne tuned. Let us now present the details of the slightly tedious construction of W
from its asymptotics.
From equation (5.2.1.6) we learn that Φ must be asymptotically constant. We denote the asymptotic values of Φ by Φ± corresponding to the limits y → ±∞. Equation
c
(5.2.1.8) implies that
∂W
= 0.
∂ Φ Φ=Φ±
c (5.2.3.24) Plugging (5.2.3.23) into (5.2.1.7) yields
W Φ+ < 0 and W Φ− > 0.
c
c (5.2.3.25) Let us look again at equation (5.2.1.8)
Φ =− 9 ∂W
,
4 ∂Φ (5.2.3.26) and view Φ as a function of Φ. Φ should reach its asymptotic values in a dynamical way
which means that Φ should be monotonically decreasing (increasing) as Φ approaches
Φ+ (Φ− ). We obtain the conditions
c
c
∂ 2W
>0 ,
∂ Φ2 Φ=Φ+
c ∂ 2W
< 0.
∂ Φ2 Φ=Φ−
c (5.2.3.27) (Equation (5.2.3.26) can be viewed as a renormalization group equation, where the
renormalization group scale is related to Φ. W is proportional to the running coupling,
and Φ (viewed as a function of Φ) is the beta function. The conditions (5.2.3.27) mean
that Φ = Φ+ (Φ = Φ− ) correspond to stable UV (IR) ﬁxed points.) Equations (5.2.1.9)
c
c
and (5.2.3.24) ﬁx the asymptotic values of the superpotential according to
V Φ− = −12W Φ−
c
c 2 , V Φ+ = −12W Φ+
c
c 2 . (5.2.3.28) This implies that the asymptotic values of V must be negative. Further note that
the asymptotic values of W are ﬁxed in a unique way with the additional conditions
(5.2.3.25). So far, we know the asymptotic value of W in terms of the input parameters
and the asymptotics of the ﬁrst derivative of W (5.2.3.24).
In order to compute the higher derivatives of W , it is useful to express the nth
derivative of V in terms of W via (5.2.1.9). The corresponding expression is
∂ nV
=
∂ Φn n 2
k =1 n − 1 ∂kW
k − 1 ∂ Φk 27 ∂ n−k+2 W
∂ n−k W
− 12
4 ∂ Φn−k+2
∂ Φn−k . (5.2.3.29) 5. The cosmological constant problem 195 This formula is most easily proven in the following way. First, apply the Leibniz rule
(F and G are arbitrary functions of Φ)
∂ n (F G)
=
∂ Φn
on ∂W 2
∂ Φ2 n n ∂ k F ∂ n−k G
k ∂ Φk ∂ Φn−k (5.2.3.30) n − 1 ∂ n−k W ∂ k W
.
k − 1 ∂ Φn−k ∂ Φk (5.2.3.31) k =0 = 2W ∂W in order to show that
∂Φ
∂nW 2
=
∂ Φn n 2
k =1 In a second step use (5.2.3.31) with W replaced by its ﬁrst derivative and redeﬁne the
summation index k → n + 1 − k. In the following we will employ (5.2.3.29) to compute the asymptotics of all derivatives of W . Since there are no singularities between the brane and the asymptotic
region, this will enable us to expand W in a Taylor series yielding its value arbitrarily
close to the brane.
The second derivative of W needs some separate discussion. With the result
(5.2.3.24) we obtain the relation
27
2 ∂ 2W
∂ Φ2 2 − 24W ∂ 2V
∂ 2W
−
∂ Φ2
∂ Φ2 = 0. (5.2.3.32) Φ=Φ±
c  This equation has real solutions for the asymptotics of the second derivative of W
provided that
8
∂ 2V
> V Φ± .
c
2 Φ=Φ±
∂Φ 
9
c (5.2.3.33) Taking into account that the asymptotic value of W is ﬁxed uniquely by (5.2.3.28)
and (5.2.3.25), and that the sign of the asymptotic value of the second derivative of
W is determined by (5.2.3.27), one ﬁnds that (5.2.3.32) can be solved in a unique way.
Note, that (5.2.3.25) and (5.2.3.27) imply
∂ 2V
> 0.
∂ Φ2 Φ=Φ±
c (5.2.3.34) The computation of the higher derivatives of W in the large y  region is somewhat
simpler. First, we notice that asymptotically on the rhs of (5.2.3.29) the nth derivative
of W is the highest occurring derivative (see (5.2.3.24)). Terms containing the nth
derivative correspond to k = 2, n. The expression (5.2.3.29) evaluated at Φ± takes the
c
form (n > 2)
∂nW
∂ nV
=
∂ Φn Φ=Φ±
∂ Φn
c 27 ∂ 2 W
− 24W
n
2 ∂ Φ2 +... Φ=Φ±
c , (5.2.3.35) 5. The cosmological constant problem 196 where the dots stand for terms containing lower derivatives of W . The relation
(5.2.3.35) allows to determine all derivatives of W provided that the coeﬃcient at
the nth derivative of W diﬀers from zero. This is ensured by equations (5.2.3.32)
and (5.2.3.34). Indeed, requiring the coeﬃcient in front of the nth derivative of W to
vanish yields
∂ 2V
32
=−
2
V ∂ Φ Φ=Φ±
9n
c 1− 1
n , (5.2.3.36) which is not compatible with (5.2.3.34) and (5.2.3.28). We conclude that in the Taylor
expansion ∞
1 ∂nW
−n n=0 n! ∂ Φn Φ=Φ− (Φ − Φc ) , y < 0
c
(5.2.3.37)
W (Φ) =
∞
1 ∂nW
+n n=0 n! ∂ Φn Φ=Φ+ (Φ − Φc ) , y > 0
c
all the coeﬃcients are ﬁxed uniquely by the model parameters. Then the jump condition (5.2.1.12) will ﬁx the value of Φ at y = 0, whereas (5.2.1.13) imposes generically
a ﬁne tuning of the model parameters.
It may look somewhat disappointing to close a review with the proof of a no go
theorem. However, often no go theorems help to ﬁnd a way leading to the desired
aim. This way should then start with a model not satisfying the prepositions of the
no go theorem. Indeed, there have been proposals for not ﬁne tuned solutions with
¯
Λ = 0. These proposals are based on the idea of introducing more integration constants
without increasing the number of jump conditions. We provide the corresponding
references in section 6. Here, we should remark that (so far) there is no commonly
accepted solution to the cosmological constant problem within brane world setups.
The explanation of the observed value of the cosmological constant remains a great
challenge. Whether branes will be helpful in a solution of this problem has to be seen
in the future. Chapter 6 Bibliography and further reading
Throughout the text I have already given some references. However, this I did only
when I felt that a direct hint on results obtained in the literature would be useful for
the reader at that particular point. Of course, these notes are based on many more
publications than already given in the text. In the present chapter, I will provide
all the references I used and give suggestions for further reading. However, there are
many more contributions to this ﬁeld. I apologize to all those authors whose work
could have been listed but is not. 6.1
6.1.1 Chapter 2
Books In [222, 223], [87], [331], [371], [269, 270] I list the textbooks on string theory of which
I am aware. In section 2.1 I used mainly[222] but also[331]. For the discussion of
orbifold planes in section 2.2 I borrowed some results presented in[371]. Dbranes and
orientifolds are also covered in[371]. Since string theory is a conformal ﬁeld theory the
book[191] may be also a useful reference. The subject of CalabiYau compactiﬁcations
entered the text rather as a side remark. Apart from the discussions presented in the
above mentioned textbooks on strings, the book[251] is perhaps a useful reference for
people who are interested in CalabiYau spaces. Let me also mention three books on
supersymmetry. Often the conventions of the standard reference[41] are used in the
literature. Ref.[456] contains (at least in its second edition) a discussion of supersymmetry in two dimensions. Finally, [399] is not really a textbook but a collection of
papers dealing with supergravity in various dimensions. For each dimension there is
a summary of the possible supermultiplets. 197 6. Review articles 6.1.2 198 Review articles Four recent review articles on perturbative string theory are [20], [368], [359] and [295].
In the context of perturbative string theory, the CFT lectures[205] may be also useful.
The computation of beta functions in nonlinear sigma models is reviewed in[445].
Various aspects of Tduality are presented in[211, 14]. Orbifold compactiﬁcations are
covered in most of the textbooks mentioned in 6.1.1. Two review articles on orbifolds
are listed in[354], [139]. K3 and other CalabiYau compactiﬁcations are e.g discussed
in[249], [224], [38], [350]. There are various reviews on Dbranes: [373, 370], [39, 40],
[437], [439], [263, 264], [130, 131]. Readers who are interested in Dbranes on CalabiYau spaces should consult[146] (and references therein). In[397], [118] two lecture notes
on orientifolds are listed. Phenomenological aspects of string theory are reviewed in
[324, 325], [380, 381], [135]. There are quite a few reviews on supersymmetry, e.g.
[355], [429], [333]. Ref. [127] presents supergravities in various dimensions. 6.1.3 Research papers For early papers on string theory I refer to the excellent commented bibliography given
in [222, 223]. Although there is still some overlap with the references in[222], I want
to start with section 2.1.3. Here, we presented details which at some points diﬀer
from the discussion in[222]. A list of references about beta functions in string sigma
models (some of them about the open string (section 2.3.3.1)) is [18], [79], [326, 327],
[189, 190], [98], [419], [99], [1], [44], [447, 446], [315], [111], [142], [49] and many others.
The normal coordinate expansion technique in section 2.1.3 is taken from[18]. In a
slightly diﬀerent version it can be found in[79]. The Fischler Susskind mechanism
is developed in [170, 171] and also discussed in e.g.[327], [48]. Like the present text,
most of the articles do not include the discussion of non trivial backgrounds for massive
string modes. The corresponding sigma model is not renormalizable. Some papers on
beta functions for massive string modes are [311], [89], [162], [314], [176], [90], [91].
Concerning section 2.1.4 I give some references related to the construction of the
supergravity theories. The existence of ten dimensional type II supergravities (and
also 11 dimensional supergravity) was suggested in[349]. The explicit construction has
been carried out in[219] (see also[411, 405, 250]). Anomalies are discussed in[21]. That
N = 1 ten dimensional supergravity coupled to E8 × E8 or SO (32) gauge theory is
anomaly free was demonstrated in[220].
Tduality for the circle compactiﬁed bosonic string is discussed in[290, 398]. For
compactiﬁcations on higher dimensional tori see[352, 425, 212]. The presentation in
section 2.1.5.3 follows closely[392]. Tduality in non trivial backgrounds with abelian
isometries was originally studied in[94, 95]. Some related papers are: [339], [213], [209], 6. Research papers 199 [296], [12, 11], [448], [207], [298]. Tduality has been also discussed for backgrounds
with non Abelian isometries e.g. in [126], [11, 13], [214], [297], [305], [160], [113, 114,
112], [17], [450], [9], [343], [242]. The Tduality relation between type IIA and type IIB
strings can be found in [136], [122], [57], [289]. The connection between compactiﬁed
E8 × E8 and SO (32) strings is presented in [206]. The techniques for orbifold compactiﬁcations of string theory have been developed in [137, 138]. More papers on orbifolds are (including explicit constructions of
phenomenological interest): [451], [237], [42], [286], [256], [257], [253], [174], [173],
[255], [318], [175]. Tduality for orbifold compactiﬁcations is for example discussed in
[312, 313].
The importance of Dbranes was realized in[369], where the connection to BPS
solutions of supergravity was discovered. In the text I have given conditions imposed
by the requirement unbroken supersymmetry on the number of ND directions. More
generally, Dbranes can intersect at certain angles[59]. The computation of Dbrane
interactions is presented in [372], [92, 93]. References concerning the beta function
approach are given together with the other references for the beta function approach
to eﬀective ﬁeld theories, above. Dbrane actions are also e.g. discussed in[400]. The
interchange of Dirichlet with Neumann boundary conditions via Tduality has been
pointed out in [122], [243], [217]. For general backgrounds, Tduality for open strings
with respect to abelian isometries is presented in[15], [143]. Tduality with respect to
nonAbelian isometries has been performed in [184], [77]. (The boundary Lagrange
multiplier has been introduced in [184, 185].) A diﬀerent method of performing Tduality transformations in general backgrounds has been proposed in [306]. The WessZumino term in the Dbrane action has been derived (in steps) in [145], [320], [61, 62],
[218]. Our discussion of open strings and non commutative geometry follows closely
(the introductory section of) [418]. Constant Bﬁelds and non commutative geometry
have been connected earlier in e.g. [147], [106], [404]. The connection between non
commutativity and the renormalization scheme is further investigated e.g. in [24, 23].
A more abstract conformal ﬁeld theory approach to Dbranes can for example be found
in [194], [388]. There are many more aspects of Dbranes for which I would like to ask
the reader to consult one of the given reviews and the references therein.
Orientifolds were introduced in[396]. For early papers on orientifold constructions
see also [378], [216], [244], [63, 64]. The cancellation of divergences in string diagrams
of type I SO (32) strings is observed in[221]. The model of section 2.4.3.2 has been
ﬁrst constructed in [63, 64]. The presentation in the text follows [204]. Indeed, it
has been the paper [204] which triggered an enormous amount of research devoted to
orientifolds. This research resulted in a lot of papers out if which I list only “a few”:
[120, 121], [203], [60], [68], [272], [469], [179], [178], [357], [5], [71, 70], [115], [377], 6. Chapter 3 200 [183], [254], [31], [25], [8], [26], [28], [27], [383], [302], [300], [301], [72], [73], [182], [382],
[74], [7, 6], [117, 116]. 6.2 Chapter 3 As far as I know there are no books devoted to solutions of ten dimensional supergravity. 6.2.1 Review articles There are quite a few review articles to be mentioned in the context of brane solutions
to supergravity. In the text I used mainly results presented in[152]. BPS solutions
to ten dimensional supergravity are also derived in [97], [431]. The theories on the
worldvolumes of the branes are discussed e.g. in [444]. Intersecting brane solutions are
e.g. reviewed in[196]. In the text I did not discuss the relevance of the brane solutions
to black hole physics. A nice introductory review to black holes is[442]. Branes in
the context of black hole physics are reviewed e.g. in[337, 338], [46], [464], [362], [428],
[151], [344, 345], [348]. 6.2.2 Research Papers The elementary string solution was found in[119]. The ﬁve brane solution has been
considered e.g. in [433], [154], [97]. The general pbrane solutions are presented in
[248]. For more references on the topic of brane solutions to supergravity I would like
to ask the reader to consult the review articles mentioned in section 6.2.1. 6.3 Chapter 4 The presented applications of branes are not a subject of a book. A discussion of
string dualities can be found in[371]. 6.3.1 Review articles There are many reviews devoted to the subject of string dualities: [441, 443], [38],
[407, 408], [452], [188], [127], [265], [423], [356], [235].
A comprehensive review on the relation between brane setups and ﬁeld theory
dualities is listed in[210]. (Another (shorter) review is [282].) In the text I mentioned
only duality relations in N = 1 supersymmetric ﬁeld theories. Such dualities are
summarized in[414, 261], [208], [365], [426]. N = 2 supersymmetric ﬁeld theories are 6. Research papers 201 considered in[65], [129], [317], [19], [33]. The duality of N = 4 super YangMills theory
is presented in [358], [239].
The standard review article on the AdS/CFT correspondence is[2]. Two more
introductory notes are listed in [366], [299]. Lecture notes dealing with Wilson loops
in the context of the AdS/CFT correspondence are e.g.[430].
Settings where the string scale is the TeV scale are reviewed in[32], [29]. 6.3.2 Research papers Early proposals of strong/weak coupling duality appear within the context of the compactiﬁed heterotic string[172], [389]. This conjecture was supported by observations
reported in[420, 422, 421], [410, 409]. The existence of 11 dimensional supergravity
was suggested in[349]. The explicit construction was carried out in[108]. The Mtheory
picture was developed in the papers[252], [440], [458]. The duality between SO (32)
type I and heterotic strings was proposed in[374]. The SL (2, Z) duality of type IIB
strings is discussed in[406]. The relation between the E8 × E8 heterotic string and
eleven dimensional supergravity is worked out in[246, 245].
Dualities in ﬁeld theories were conjectured in[346], and shown to be exact in N = 4
supersymmetric Yang Mills theory in[463], [360], [457]. Strong coupling results in
N = 2 gauge theories are presented in [416], [417], for SU (2). Extensions to other
gauge groups are discussed in e.g.[304], [303], [34], [167], [166]. The N = 1 ﬁeld
theory dualities have been conjectured in[412, 413]. Some out of many subsequent
papers are[316], [258], [260], [259], [156, 157], [309], [310], [88]. Studying ﬁeld theories
via manipulations in brane setups was initiated in[238]. The discussion in the text
follows[158]. There are many related works. Some examples are: [125], [159], [165],
[82], [3], [436], [283]. The connection between N = 2 supersymmetric gauge theories
and Mtheory branes is considered in[459]. There is also a larger list of literature
dealing with brane setups for N = 2 theories, for which, however, I would like to ask
the reader to consult one of the reviews since this would lead to far away from the
subjects discussed in the text.
The AdS/CFT correspondence is conjectured in[335], and further elaborated in[233],
[460]. The computation of Wilson loops within the conjecture is described in[336],
[391]. Diﬀerently shaped Wilson loops are discussed in[54], [148]. Breaking supersymmetry by a ﬁnite temperature one can observe the conﬁnement of quarks[461]. Related
papers are[81], [390], [80], [230], [234], [144], [361] and many others. The string action on AdS5 × S 5 is constructed in[340]. This action is discussed further in[276], [275], [273], [364], [277], [384]. The construction of[340] leads also to the result that
the AdS5 × S 5 background is exact. Diﬀerent arguments for this statement are given 6. Chapter 5 202 in[43]. The discussion of the stringy corrections to the Wilson loop follows[180, 181].
A similar approach (in the conformal gauge) and more examples are discussed in[149].
This paper also addresses the problem of the divergence and gives a numerical estimate
of the correction. String ﬂuctuations as a source for corrections to the Wilson loop are
also discussed in[226], [351], [466], [294], [262], [322]. Corrections to the ﬁeld theory
calculation are derived in[163], [467], [164],[367]. An attempt to apply the techniques
for computing corrections to the Wilson loop on the M5 brane case is reported in[177].
That branes allow constructions with the string scale at a TeV has been pointed
out in[30]. (Relating the hierarchy problem to the size of extra dimensions has been
proposed before in a ﬁeld theory context[35].) The argument that in compactiﬁcations
of the perturbative heterotic string the size of the compact space is of the order of the
Planck size is given in[281]. Our discussion of corrections to Newton’s law follows[287]. 6.4 Chapter 5 Since there are no books on the subject of brane world setups I start directly with a
list of review articles. 6.4.1 Review articles The review articles on brane world setups with warped transverse dimensions I am
aware of are[266], [393], [133], [334]. An overview on the cosmological constant problem
is presented in[455], [462], [66]. 6.4.2 Research papers Brane world models have been proposed already sometime back in[394], [4]. The model
discussed in section 5.1.1 is presented in[386]. The stabilization mechanism is proposed
in[215]. The model of section 5.1.2 is taken from[385]. An early paper on connecting
the Randall Sundrum model with the holographic principle is[454]. The computation
of the Newton potential via the holographic principle has been pointed out by Witten
in the discussion session in a Santa Barbara Conference in 1999. (I have not been
there.) The presentation in the text is taken from[232] (see also[153]). The inclusion
of the second brane into the RS2 scenario is performed in[332]. The computation
of the Newton potential via the AdS/CFT correspondence is taken from[37] (see also
[201]). More discussions of the RS models from a holographic perspective can be found
e.g. in[22], [78], [387], [363] , [284, 285], [427], [197], [107]. Supersymmetry within the
context of the Randall Sundrum model is discussed in[274], [47], [10], [198], [168, 169],
[58], [468]. 6. Research papers 203 Section 5.2.1 is closely related to[128]. The consistency condition that the eﬀective cosmological constant should be compatible with the metric on the brane is also
mentioned in[128]. The derivation and form of the consistency condition in section
5.2.2 is presented in[187]. The alternative method of integrating a total derivative is
developed in[161]. The connection between the two conditions has been pointed out
in[187]. The complete set of consistency conditions (as it appears in the text) is given
in[199]. (Diﬀerent consistency conditions are discussed in[279].)
That the cosmological constant problem is rephrased within a brane world setup
is discussed in[395]. The example of section 5.2.3.1 (and a closely related example)
appear in[268], [36]. That the eﬀective cosmological constant does not vanish in this
models is observed (simultaneously) in [465], [186]. To reach consistency by adding
branes and consequently ﬁne tuning input parameters is proposed in [186]. (Problems
with singularities in warped compactiﬁcations are considered e.g. also in[231], [291].)
The proof of the no go theorem is taken from[110].
There are too many papers on warped brane world scenarios to be listed. Therefore,
the following list is restricted to papers dealing with the cosmological constant problem
(and most likely this list is also incomplete): [101], [123], [280], [124], [247], [227], [45],
[308, 307], [105], [67], [321], [271], [278], [104], [55], [85], [100], [103], [134], [86], [240].
Papers containing proposals on avoiding the ﬁne tuning problem of the cosmological
constant by going beyond the prepositions of the no go theorem (section 5.2.3.2)
are [288], [292, 293], [109, 229]. A diﬀerent proposal for addressing the cosmological
constant problem in brane world scenarios is put forward in[453], [401, 402, 403].
Warped compactiﬁcations in the context of string theory are e.g. discussed in[330,
329], [53], [150], [69], [84].
Observational bounds on extra dimension scenarios are e.g. presented in[323], [342],
[341]. Acknowledgments
The topics of these notes represent the experience with string theory that the author
gained over the past years. During this period I collaborated with and received invaluable help from various people. First, I thank HansJ¨rg Otto and Harald Dorn
o
who patiently taught me the basics of string theory, the beta function approach to
the eﬀective ﬁeld theory, and non critical strings. I enjoyed very much collaborating
with Klaus Behrndt on[50, 51, 52]. These collaborations have been very eﬃcient even
at times when we communicated only via email. I would like to thank the Jerusalem
group: Shmuel Elitzur, Amit Giveon and Eliezer Rabinovici for many insights into the
subjects of Tduality, marginal deformations of conformal ﬁeld theories, and strongly 6. Research papers 204 coupled supersymmetric gauge theories. I also acknowledge many enjoyable discussions
with Gautam Sengupta. In Munich, I learned quite a lot about string dualities from
Jan Louis and Stefan Theisen. I would like to thank Stefan Schwager and Alexandros
Kehagias for collaborations on Tduality in open string models. I had many interesting discussions with Kristin F¨rger, Debashis Ghoshal, Jacek Pawelczyk and Emanuel
o
Scheidegger. I thank Debashis Ghoshal and Sudhakar Panda for collaborating on
orientifold constructions. I also enjoyed very much the collaboration with Debashis
Ghoshal and Stefan Theisen on stringy corrections to the Wilson loop in the context
of the AdS/CFT correspondence. Finally, it is my pleassure to express my gratitude
to the high energy physics group in Bonn. I acknowledge the fruitful collaboration
with Zygmunt Lalak, St´phane Lavignac and HansPeter Nilles on the cosmological
e
constant problem. Many thanks to Gabriele Honecker and Ralph Schreyer for collaborating with me on orientifolds. While writing these notes I had many helpful
discussions with Jan Conrad, Athanasios Dedes, Dumitru Ghilencea, Stefan Groot
Nibbelink, Gabriele Honecker, Mark Hillenbach, Hanno Klemm, Marco Peloso, Ralph
Schreyer and Martin Walter. Special thanks to Gabriele Honecker and Ralph Schreyer
for frequent proof reading of the manuscript. Part of these notes I presented in a lecture
during the summer term 2001. I thank the students for the stimulating atmosphere
during these lectures and Martin Walter for preparing and delivering accompanying
exercises. Last but not least, I would like to express my gratitude to HansPeter Nilles
for his steady support and encouragement.
My current work is supported in part by the European Commission RTN programs
HPRNCT200000131, 00148 and 00152. Bibliography
[1] A. Abouelsaood, C. G. Callan, C. R. Nappi, and S. A. Yost. Open strings in
background gauge ﬁelds. Nucl. Phys., B280:599, 1987.
[2] O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz. Large N ﬁeld theories, string theory and gravity. Phys. Rept., 323:183–386, 2000. hepth/9905111.
[3] O. Aharony and A. Hanany. Branes, superpotentials and superconformal ﬁxed
points. Nucl. Phys., B504:239–271, 1997. hepth/9704170.
[4] K. Akama. An early proposal of ’brane world’. Lect. Notes Phys., 176:267–271,
1982. hepth/0001113.
[5] G. Aldazabal, A. Font, L. E. Ib´nez, and G. Violero. D = 4, N = 1, type IIB
a˜
orientifolds. Nucl. Phys., B536:29–68, 1998. hepth/9804026.
[6] G. Aldazabal, S. Franco, L. E. Ib´nez, R. Rabadan, and A. M. Uranga. D = 4
a˜
chiral string compactiﬁcations from intersecting branes. 2000. hepth/0011073.
[7] G. Aldazabal, S. Franco, L. E. Ib´nez, R. Rabadan, and A. M. Uranga. Intera˜
secting brane worlds. JHEP, 02:047, 2001. hepph/0011132.
[8] G. Aldazabal and A. M. Uranga. Tachyonfree nonsupersymmetric type IIB
orientifolds via braneantibrane systems. JHEP, 10:024, 1999. hepth/9908072.
[9] A. Yu. Alekseev, C. Klimˇik, and A. A. Tseytlin. Quantum PoissonLie Tduality
c
and WZNW model. Nucl. Phys., B458:430–444, 1996. hepth/9509123.
[10] R. Altendorfer, J. Bagger, and D. Nemeschansky. Supersymmetric RandallSundrum scenario. Phys. Rev., D63:125025, 2001. hepth/0003117.
[11] E. Alvarez, L. AlvarezGaum´, J. L. F. Barb´n, and Y. Lozano. Some global
e
o
aspects of duality in string theory.
th/9309039. 205 Nucl. Phys., B415:71–100, 1994. hep BIBLIOGRAPHY 206 [12] E. Alvarez, L. AlvarezGaum´, and Y. Lozano. A Canonical approach to duality
e
transformations. Phys. Lett., B336:183–189, 1994. hepth/9406206.
[13] E. Alvarez, L. AlvarezGaum´, and Y. Lozano. On nonAbelian duality. Nucl.
e
Phys., B424:155–183, 1994. hepth/9403155.
[14] E. Alvarez, L. AlvarezGaum´, and Y. Lozano. An introduction to T duality in
e
string theory. Nucl. Phys. Proc. Suppl., 41:1–20, 1995. hepth/9410237.
[15] E. Alvarez, J. L. F. Barb´n, and J. Borlaf. Tduality for open strings. Nucl.
o
Phys., B479:218–242, 1996. hepth/9603089.
[16] O. Alvarez. Theory of strings with boundaries: Fluctuations, topology, and
quantum geometry. Nucl. Phys., B216:125, 1983.
[17] O. Alvarez and C. Liu. Target space duality between simple compact Lie groups
and Lie algebras under the Hamiltonian formalism: 1. Remnants of duality at
the classical level. Commun. Math. Phys., 179:185–214, 1996. hepth/9503226.
[18] L. AlvarezGaume, D. Z. Freedman, and S. Mukhi. The background ﬁeld method
and the ultraviolet structure of the supersymmetric nonlinear sigma model. Ann.
Phys., 134:85, 1981.
[19] L. AlvarezGaum´ and S. F. Hassan. Introduction to Sduality in N = 2 sue
persymmetric gauge theories: A pedagogical review of the work of Seiberg and
Witten. Fortsch. Phys., 45:159–236, 1997. hepth/9701069.
[20] L. AlvarezGaum´ and M. A. VazquezMozo. Topics in string theory and quane
tum gravity. 1992. hepth/9212006.
[21] L. AlvarezGaum´ and E. Witten.
e
B234:269, 1984. Gravitational anomalies. Nucl. Phys., [22] L. Anchordoqui, C. Nunez, and K. Olsen. Quantum cosmology and AdS/CFT.
JHEP, 10:050, 2000. arXiv:hepth/0007064.
[23] O. Andreev and H. Dorn. Diagrams of noncommutative Φ3 theory from string
theory. Nucl. Phys., B583:145–158, 2000. hepth/0003113.
[24] O. Andreev and H. Dorn. On open string sigmamodel and noncommutative
gauge ﬁelds. Phys. Lett., B476:402–410, 2000. hepth/9912070.
[25] C. Angelantonj. Comments on openstring orbifolds with a nonvanishing Bab .
Nucl. Phys., B566:126–150, 2000. hepth/9908064. BIBLIOGRAPHY 207 [26] C. Angelantonj, I. Antoniadis, G. D’Appollonio, E. Dudas, and A. Sagnotti.
Type I vacua with brane supersymmetry breaking. Nucl. Phys., B572:36–70,
2000. hepth/9911081.
[27] C. Angelantonj, I. Antoniadis, E. Dudas, and A. Sagnotti. TypeI strings on
magnetised orbifolds and brane transmutation. Phys. Lett., B489:223–232, 2000.
hepth/0007090.
[28] C. Angelantonj, R. Blumenhagen, and M. R. Gaberdiel. Asymmetric orientifolds,
brane supersymmetry breaking and nonBPS branes. Nucl. Phys., B589:545–576,
2000. hepth/0006033.
[29] I. Antoniadis. String and Dbrane physics at low energy. 2001. hepth/0102202.
[30] I. Antoniadis, N. ArkaniHamed, S. Dimopoulos, and G. Dvali. New dimensions
at a millimeter to a Fermi and superstrings at a TeV. Phys. Lett., B436:257–263,
1998. hepph/9804398.
[31] I. Antoniadis, E. Dudas, and A. Sagnotti. Supersymmetry breaking, open strings
and Mtheory. Nucl. Phys., B544:469–502, 1999. hepth/9807011.
[32] I. Antoniadis and A. Sagnotti. Mass scales, supersymmetry breaking and open
strings. Class. Quant. Grav., 17:939–950, 2000. hepth/9911205.
[33] P. C. Argyres. Dualities in supersymmetric ﬁeld theories. Nucl. Phys. Proc.
Suppl., 61A:149–157, 1998. hepth/9705076.
[34] P. C. Argyres and M. R. Douglas. New phenomena in SU(3) supersymmetric
gauge theory. Nucl. Phys., B448:93–126, 1995. hepth/9505062.
[35] N. ArkaniHamed, S. Dimopoulos, and G. Dvali. The hierarchy problem and new
dimensions at a millimeter. Phys. Lett., B429:263–272, 1998. hepph/9803315.
[36] N. ArkaniHamed, S. Dimopoulos, N. Kaloper, and R. Sundrum. A small cosmological constant from a large extra dimension. Phys. Lett., B480:193–199, 2000.
hepth/0001197.
[37] N. ArkaniHamed, M. Porrati, and L. Randall. Holography and phenomenology.
JHEP, 08:017, 2001. hepth/0012148.
[38] P. S. Aspinwall. K3 surfaces and string duality. 1996. hepth/9611137.
[39] C. Bachas. (half) a lecture on Dbranes. 1996. hepth/9701019.
[40] C. P. Bachas. Lectures on Dbranes. 1998. hepth/9806199. BIBLIOGRAPHY 208 [41] J. Bagger and J. Wess. Supersymmetry and supergravity. JHUTIPAC9009.
[42] J. A. Bagger, Jr. Callan, C. G., and J. A. Harvey. Cosmic strings as orbifolds.
Nucl. Phys., B278:550, 1986.
[43] T. Banks and M. B. Green. Nonperturbative eﬀects in AdS5 × S 5 string theory
and d = 4 susy YangMills. JHEP, 05:002, 1998. hepth/9804170.
[44] T. Banks and E. Martinec. The renormalization group and string ﬁeld theory.
Nucl. Phys., B294:733, 1987.
[45] V. Barger, T. Han, T. Li, J. D. Lykken, and D. Marfatia. Cosmology and
hierarchy in stabilized warped brane models. Phys. Lett., B488:97–107, 2000.
hepph/0006275.
[46] K. Behrndt. Classical and quantum aspects of 4 dimensional black holes. Nucl.
Phys. Proc. Suppl., 56B:11–22, 1997. hepth/9701053.
[47] K. Behrndt and M. Cvetiˇ. Antide Sitter vacua of gauged supergravities with
c
8 supercharges. Phys. Rev., D61:101901, 2000. hepth/0001159.
[48] K. Behrndt and H. Dorn. Boundary eﬀects relevant for the string interpretation
of sigma models. Phys. Lett., B266:59–64, 1991.
[49] K. Behrndt and H. Dorn. Weyl anomaly and CurciPaﬀuti theorem for sigma
models on manifolds with boundary. Int. J. Mod. Phys., A7:1375–1390, 1992.
[50] K. Behrndt and S. F¨rste. Cosmological string solutions in fourdimensions from
o
5d black holes. Phys. Lett., B320:253–258, 1994. hepth/9308131.
[51] K. Behrndt and S. F¨rste. String KaluzaKlein cosmology.
o
B430:441–459, 1994. hepth/9403179. Nucl. Phys., [52] K. Behrndt, S. F¨rste, and S. Schwager. Instanton eﬀects in string cosmology.
o
Nucl. Phys., B508:391–408, 1997. hepth/9704013.
[53] K. Behrndt and S. Gukov. Domain walls and superpotentials from M theory on
CalabiYau threefolds. Nucl. Phys., B580:225–242, 2000. hepth/0001082.
[54] D. Berenstein, R. Corrado, W. Fischler, and J. Maldacena. The operator product expansion for Wilson loops and surfaces in the large N limit. Phys. Rev.,
D59:105023, 1999. hepth/9809188. BIBLIOGRAPHY 209 [55] Z. Berezhiani, M. Chaichian, A. B. Kobakhidze, and Z. H. Yu. Vanishing of
cosmological constant and fully localized gravity in a brane world with extra
time(s). 2001. hepth/0102207.
[56] E. Bergshoeﬀ, M. de Roo, M. B. Green, G. Papadopoulos, and P. K. Townsend.
Duality of Type II 7branes and 8branes. Nucl. Phys., B470:113–135, 1996.
hepth/9601150.
[57] E. Bergshoeﬀ, C. Hull, and T. Ortin. Duality in the type II superstring eﬀective
action. Nucl. Phys., B451:547–578, 1995. hepth/9504081.
[58] E. Bergshoeﬀ, R. Kallosh, and A. Van Proeyen. Supersymmetry in singular
spaces. JHEP, 10:033, 2000. hepth/0007044.
[59] M. Berkooz, M. R. Douglas, and R. G. Leigh. Branes intersecting at angles.
Nucl. Phys., B480:265–278, 1996. hepth/9606139.
[60] M. Berkooz and R. G. Leigh. A D = 4 N = 1 orbifold of type I strings. Nucl.
Phys., B483:187–208, 1997. hepth/9605049.
[61] M. Bershadsky, C. Vafa, and V. Sadov. DBranes and Topological Field Theories.
Nucl. Phys., B463:420–434, 1996. hepth/9511222.
[62] M. Bershadsky, C. Vafa, and V. Sadov. DStrings on DManifolds. Nucl. Phys.,
B463:398–414, 1996. hepth/9510225.
[63] M. Bianchi and A. Sagnotti. On the systematics of open string theories. Phys.
Lett., B247:517–524, 1990.
[64] M. Bianchi and A. Sagnotti. Twist symmetry and open string Wilson lines.
Nucl. Phys., B361:519–538, 1991.
[65] A. Bilal. Duality in N=2 SUSY SU(2) YangMills Theory: A pedagogical introduction to the work of seiberg and witten. 1995. hepth/9601007.
[66] P. Binetruy. Cosmological constant vs. quintessence. Int. J. Theor. Phys.,
39:1859–1875, 2000. hepph/0005037.
[67] P. Binetruy, J. M. Cline, and C. Grojean. Dynamical instability of brane solutions with a selftuning cosmological constant. Phys. Lett., B489:403–410, 2000.
hepth/0007029.
[68] J. D. Blum and A. Zaﬀaroni. An orientifold from F theory. Phys. Lett., B387:71–
74, 1996. hepth/9607019. BIBLIOGRAPHY 210 [69] R. Blumenhagen and A. Font. Dilaton tadpoles, warped geometries and large
extra dimensions for nonsupersymmetric strings. Nucl. Phys., B599:241–254,
2001. hepth/0011269.
[70] R. Blumenhagen, L. G¨rlich, and B. K¨rs. Supersymmetric 4D orientifolds of
o
o
type IIA with D6branes at angles. JHEP, 01:040, 2000. hepth/9912204.
[71] R. Blumenhagen, L. G¨rlich, and B. K¨rs. Supersymmetric orientifolds in 6D
o
o
with Dbranes at angles. Nucl. Phys., B569:209–228, 2000. hepth/9908130.
[72] R. Blumenhagen, L. G¨rlich, B. K¨rs, and D. L¨st. Magnetic ﬂux in toroidal
o
o
u
type I compactiﬁcation. Fortsch. Phys., 49:591–598, 2001. hepth/0010198.
[73] R. Blumenhagen, B. K¨rs, and D. L¨st. Type I strings with F and Bﬂux.
o
u
JHEP, 02:030, 2001. hepth/0012156.
[74] R. Blumenhagen, B. K¨rs, D. L¨st, and T. Ott. The standard model from stable
o
u
intersecting brane world orbifolds. 2001. hepth/0107138.
[75] J. Bogaerts, A. Sevrin, J. Troost, W. Troost, and S. van der Loo. Dbranes and
constant electromagnetic backgrounds. 2000. hepth/0101018.
[76] E. B. Bogomolny. Stability of classical solutions. Sov. J. Nucl. Phys., 24:449,
1976.
[77] J. Borlaf and Y. Lozano. Aspects of Tduality in open strings. Nucl. Phys.,
B480:239–264, 1996. hepth/9607051.
[78] H. BoschiFilho and N. R. F. Braga. Field spectrum and degrees of freedom in
AdS/CFT correspondence and Randall Sundrum model. Nucl. Phys., B608:319–
332, 2001. arXiv:hepth/0012196.
[79] E. Braaten, T. L. Curtright, and C. Zachos. Torsion and geometrostasis in
nonlinear sigma models. Nucl. Phys., B260:630, 1985.
[80] A. Brandhuber, N. Itzhaki, J. Sonnenschein, and S. Yankielowicz. Wilson loops,
conﬁnement, and phase transitions in large N gauge theories from supergravity.
JHEP, 06:001, 1998. hepth/9803263.
[81] A. Brandhuber, N. Itzhaki, J. Sonnenschein, and S. Yankielowicz. Wilson loops
in the large N limit at ﬁnite temperature. Phys. Lett., B434:36–40, 1998. hepth/9803137. BIBLIOGRAPHY 211 [82] A. Brandhuber, J. Sonnenschein, S. Theisen, and S. Yankielowicz. Brane conﬁgurations and 4D ﬁeld theory dualities. Nucl. Phys., B502:125–148, 1997. hepth/9704044.
[83] T. Branson, P. Gilkey, and D. Vassilevich. Vacuum expectation value asymptotics for second order diﬀerential operators on manifolds with boundary. J.
Math. Phys., 39:1040–1049, 1998. hepth/9702178.
[84] P. Brax. NonBPS instability in heterotic Mtheory. Phys. Lett., B506:362–368,
2001. hepth/0102154.
[85] P. Brax and A. C. Davis. Cosmological evolution on selftuned branes and the
cosmological constant. JHEP, 05:007, 2001. hepth/0104023.
[86] P. Brax and A. C. Davis. On brane cosmology and naked singularities. Phys.
Lett., B513:156–162, 2001. hepth/0105269.
[87] L. Brink and M. Henneaux. Principles of string theory. NEW YORK, USA:
PLENUM (1988) 297p.
[88] J. H. Brodie. Duality in supersymmetric SU (Nc ) gauge theory with two adjoint
chiral superﬁelds. Nucl. Phys., B478:123–140, 1996. hepth/9605232.
[89] R. Brustein, D. Nemeschansky, and S. Yankielowicz. Beta functions and S matrix
in string theory. Nucl. Phys., B301:224, 1988.
[90] I. L. Buchbinder, E. S. Fradkin, S. L. Lyakhovich, and V. D. Pershin. Higher
spins dynamics in the closed string theory. Phys. Lett., B304:239–248, 1993.
hepth/9304131.
[91] I. L. Buchbinder, V. A. Krykhtin, and V. D. Pershin. Massive ﬁeld dynamics in
open bosonic string theory. Phys. Lett., B348:63–69, 1995. hepth/9412132.
[92] C. P. Burgess and T. R. Morris. Open and unoriented strings a la Polyakov.
Nucl. Phys., B291:256, 1987.
[93] C. P. Burgess and T. R. Morris. Open superstrings a la Polyakov. Nucl. Phys.,
B291:285, 1987.
[94] T. H. Buscher. Quantum corrections and extended supersymmetry in new sigma
models. Phys. Lett., B159:127, 1985.
[95] T. H. Buscher. A symmetry of the string background ﬁeld equations. Phys.
Lett., B194:59, 1987. BIBLIOGRAPHY 212 [96] Jr. C. G. Callan, J. A. Harvey, and A. Strominger. Supersymmetric string
solitons. 1991. hepth/9112030.
[97] Jr. C. G. Callan, J. A. Harvey, and A. Strominger. World sheet approach to
heterotic instantons and solitons. Nucl. Phys., B359:611–634, 1991.
[98] C. G. Callan, E. J. Martinec, M. J. Perry, and D. Friedan. Strings in background
ﬁelds. Nucl. Phys., B262:593, 1985.
[99] P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten. Vacuum conﬁgurations for superstrings. Nucl. Phys., B258:46–74, 1985.
[100] S. M. Carroll and L. Mersini. Can we live in a selftuning universe?
hepth/0105007. 2001. [101] J. Chen, M. A. Luty, and E. Ponton. A critical cosmological constant from
millimeter extra dimensions. JHEP, 09:012, 2000. hepth/0003067.
[102] K. G. Chetyrkin and F. V. Tkachov. Infrared R operation and ultraviolet counterterms in the MS scheme. Phys. Lett., B114:340–344, 1982.
[103] J. M. Cline. Quintessence, cosmological horizons, and selftuning. JHEP, 08:035,
2001. hepph/0105251.
[104] J. M. Cline and H. Firouzjahi. A small cosmological constant from warped
compactiﬁcation with branes. Phys. Lett., B514:205–212, 2001. hepph/0012090.
[105] H. Collins and B. Holdom. The cosmological constant and warped extra dimensions. Phys. Rev., D63:084020, 2001. hepth/0009127.
[106] A. Connes, M. R. Douglas, and A. Schwarz. Noncommutative geometry and
matrix theory: Compactiﬁcation on tori. JHEP, 02:003, 1998. hepth/9711162.
[107] P. Creminelli, A. Nicolis, and R. Rattazzi. Holography and the electroweak phase
transition. 2001. arXiv:hepth/0107141.
[108] E. Cremmer, B. Julia, and J. Scherk. Supergravity theory in 11 dimensions.
Phys. Lett., B76:409–412, 1978.
[109] C. Csaki, J. Erlich, and C. Grojean. Gravitational Lorentz violations and adjustment of the cosmological constant in asymmetrically warped spacetimes. Nucl.
Phys., B604:312–342, 2001. hepth/0012143. BIBLIOGRAPHY 213 [110] C. Csaki, J. Erlich, C. Grojean, and T. Hollowood. General properties of the
selftuning domain wall approach to the cosmological constant problem. Nucl.
Phys., B584:359–386, 2000. hepth/0004133.
[111] G. Curci and G. Paﬀuti. Consistency between the string background ﬁeld
equation of motion and the vanishing of the conformal anomaly. Nucl. Phys.,
B286:399, 1987.
[112] T. Curtright, T. Uematsu, and C. Zachos. Geometry and Duality in Supersymmetric sigmaModels. Nucl. Phys., B469:488–512, 1996. hepth/9601096.
[113] T. Curtright and C. Zachos. Currents, charges, and canonical structure of pseudodual chiral models. Phys. Rev., D49:5408–5421, 1994. hepth/9401006.
[114] T. Curtright and C. Zachos. Canonical nonAbelian dual transformations in
supersymmetric ﬁeld theories. Phys. Rev., D52:573–576, 1995. hepth/9502126.
[115] M. Cvetiˇ, M. Plumacher, and J. Wang. Three family type IIB orientifold string
c
vacua with nonAbelian wilson lines. JHEP, 04:004, 2000. hepth/9911021.
[116] M. Cvetiˇ, G. Shiu, and A. M. Uranga. Chiral fourdimensional N = 1 supersymc
metric type IIA orientifolds from intersecting D6branes. 2001. hepth/0107166.
[117] M. Cvetiˇ, G. Shiu, and A. M. Uranga. Threefamily supersymmetric standard
c
like models from intersecting brane worlds. 2001. hepth/0107143.
[118] A. Dabholkar. Lectures on orientifolds and duality. 1997. hepth/9804208.
[119] A. Dabholkar, G. Gibbons, J. A. Harvey, and F. Ruiz Ruiz. Superstrings and
solitons. Nucl. Phys., B340:33–55, 1990.
[120] A. Dabholkar and J. Park. An Orientifold of TypeIIB Theory on K 3. Nucl.
Phys., B472:207–220, 1996. hepth/9602030.
[121] A. Dabholkar and J. Park. Strings on orientifolds. Nucl. Phys., B477:701–714,
1996. hepth/9604178.
[122] J. Dai, R. G. Leigh, and J. Polchinski. New connections between string theories.
Mod. Phys. Lett., A4:2073–2083, 1989.
[123] S. P. de Alwis. Brane world scenarios and the cosmological constant. Nucl.
Phys., B597:263–278, 2001. hepth/0002174.
[124] S. P. de Alwis, A. T. Flournoy, and N. Irges. Brane worlds, the cosmological
constant and string theory. JHEP, 01:027, 2001. hepth/0004125. BIBLIOGRAPHY 214 [125] J. de Boer, K. Hori, H. Ooguri, Y. Oz, and Z. Yin. Mirror symmetry in threedimensional gauge theories, SL(2, Z) and Dbrane moduli spaces. Nucl. Phys.,
B493:148–176, 1997. hepth/9612131.
[126] X. C. de la Ossa and F. Quevedo. Duality symmetries from nonAbelian isometries
in string theory. Nucl. Phys., B403:377–394, 1993. hepth/9210021.
[127] B. de Wit and J. Louis. Supersymmetry and dualities in various dimensions.
1997. hepth/9801132.
[128] O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch. Modeling the ﬁfth dimension with scalars and gravity. Phys. Rev., D62:046008, 2000. hepth/9909134.
[129] P. Di Vecchia. Duality in supersymmetric gauge theories. Surveys High Energ.
Phys., 10:119, 1997. hepth/9608090.
[130] P. Di Vecchia and A. Liccardo. D branes in string theory. I. 1999. hepth/9912161.
[131] P. Di Vecchia and A. Liccardo. Dbranes in string theory. II. 1999. hepth/9912275.
[132] R. Dick. Which action for brane worlds? Phys. Lett., B491:333–338, 2000. hepth/0007063.
[133] R. Dick. Brane worlds. 2001. hepth/0105320.
[134] J. Diemand, C. Mathys, and D. Wyler. Dynamical instabilities of brane world
models. 2001. hepth/0105240.
[135] K. R. Dienes. String Theory and the Path to Uniﬁcation: A Review of Recent
Developments. Phys. Rept., 287:447–525, 1997. hepth/9602045.
[136] M. Dine, P. Huet, and N. Seiberg. Large and small radius in string theory. Nucl.
Phys., B322:301, 1989.
[137] L. Dixon, J. A. Harvey, C. Vafa, and E. Witten. Strings on orbifolds. Nucl.
Phys., B261:678–686, 1985.
[138] L. Dixon, J. A. Harvey, C. Vafa, and E. Witten. Strings on orbifolds. 2. Nucl.
Phys., B274:285–314, 1986.
[139] L. J. Dixon. Some world sheet properties of superstring compactiﬁcations, on
orbifolds and otherwise. Lectures given at the 1987 ICTP Summer Workshop in
High Energy Phsyics and Cosmology, Trieste, Italy, Jun 29  Aug 7, 1987. BIBLIOGRAPHY 215 [140] H. Dorn. Nonabelian gauge ﬁeld dynamics on matrix Dbranes. Nucl. Phys.,
B494:105–118, 1997. hepth/9612120.
[141] H. Dorn. The mass term in nonAbelian gauge ﬁeld dynamics on matrix Dbranes and Tduality in the sigmamodel approach. JHEP, 04:013, 1998. hepth/9804065.
[142] H. Dorn and H. J. Otto. Open bosonic strings in general background ﬁelds. Z.
Phys., C32:599, 1986.
[143] H. Dorn and H. J. Otto. On Tduality for open strings in general abelian and nonabelian gauge ﬁeld backgrounds. Phys. Lett., B381:81–88, 1996. hepth/9603186.
[144] H. Dorn and H. J. Otto. On Wilson loops and Q antiQ potentials from the
AdS/CFT relation at T ≥ 0. 1998. hepth/9812109.
[145] M. R. Douglas. Branes within branes. 1995. hepth/9512077.
[146] M. R. Douglas. Dbranes on CalabiYau manifolds. 2000. math.ag/0009209.
[147] M. R. Douglas and C. Hull. Dbranes and the noncommutative torus. JHEP,
02:008, 1998. hepth/9711165.
[148] N. Drukker, D. J. Gross, and H. Ooguri. Wilson loops and minimal surfaces.
Phys. Rev., D60:125006, 1999. hepth/9904191.
[149] N. Drukker, D. J. Gross, and A. A. Tseytlin. GreenSchwarz string in AdS5 × S 5 :
Semiclassical partition function. JHEP, 04:021, 2000. hepth/0001204. [150] E. Dudas and J. Mourad. Brane solutions in strings with broken supersymmetry
and dilaton tadpoles. Phys. Lett., B486:172–178, 2000. hepth/0004165.
[151] M. J. Duﬀ. TASI lectures on branes, black holes and antide Sitter space. 1999.
hepth/9912164.
[152] M. J. Duﬀ, Ramzi R. Khuri, and J. X. Lu. String solitons. Phys. Rept., 259:213–
326, 1995. hepth/9412184.
[153] M. J. Duﬀ and James T. Liu. Complementarity of the Maldacena and RandallSundrum pictures. Phys. Rev. Lett., 85:2052–2055, 2000. hepth/0003237.
[154] M. J. Duﬀ and J. X. Lu. Elementary ﬁvebrane solutions of D = 10 supergravity.
Nucl. Phys., B354:141–153, 1991. BIBLIOGRAPHY 216 [155] L. P. Eisenhart. Riemannian Geometry. Princeton University Press, Princeton
(1964).
[156] S. Elitzur, A. Forge, A. Giveon, and E. Rabinovici. More results in N=1 supersymmetric gauge theories. Phys. Lett., B353:79–83, 1995. hepth/9504080.
[157] S. Elitzur, A. Forge, A. Giveon, and E. Rabinovici. Eﬀective Potentials and Vacuum Structure in N=1 Supersymmetric Gauge Theories. Nucl. Phys., B459:160–
206, 1996. hepth/9509130.
[158] S. Elitzur, A. Giveon, and D. Kutasov. Branes and N = 1 duality in string
theory. Phys. Lett., B400:269–274, 1997. hepth/9702014.
[159] S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici, and A. Schwimmer. Brane
dynamics and N = 1 supersymmetric gauge theory. Nucl. Phys., B505:202–250,
1997. hepth/9704104.
[160] S. Elitzur, A. Giveon, E. Rabinovici, A. Schwimmer, and G. Veneziano. Remarks
on nonAbelian duality. Nucl. Phys., B435:147–171, 1995. hepth/9409011.
[161] U. Ellwanger. Constraints on a braneworld from the vanishing of the cosmological constant. Phys. Lett., B473:233–240, 2000. hepth/9909103.
[162] U. Ellwanger and J. Fuchs. String ﬁeld equations and the dual S matrix from
the exact renormalization group. Nucl. Phys., B312:95, 1989.
[163] J. K. Erickson, G. W. Semenoﬀ, R. J. Szabo, and K. Zarembo. Static potential
in N = 4 supersymmetric YangMills theory. Phys. Rev., D61:105006, 2000.
hepth/9911088.
[164] J. K. Erickson, G. W. Semenoﬀ, and K. Zarembo. Wilson loops in N = 4 supersymmetric YangMills theory. Nucl. Phys., B582:155–175, 2000. hepth/0003055.
[165] N. Evans, C. V. Johnson, and A. D. Shapere. Orientifolds, branes, and duality
of 4D gauge theories. Nucl. Phys., B505:251–271, 1997. hepth/9703210.
[166] H. Ewen and K. F¨rger. Simple calculation of instanton corrections in massive
o
N = 2 SU(3) SYM. Int. J. Mod. Phys., A12:4725–4742, 1997. hepth/9610049.
[167] H. Ewen, K. F¨rger, and S. Theisen. Prepotentials in N = 2 supersymmetric
o
SU(3) YM theory with massless hypermultiplets. Nucl. Phys., B485:63–84, 1997.
hepth/9609062. BIBLIOGRAPHY 217 [168] A. Falkowski, Z. Lalak, and S. Pokorski. Supersymmetrizing branes with bulk in
ﬁvedimensional supergravity. Phys. Lett., B491:172–182, 2000. hepth/0004093.
[169] A. Falkowski, Z. Lalak, and S. Pokorski. Fivedimensional gauged supergravities
with universal hypermultiplet and warped brane worlds. Phys. Lett., B509:337–
345, 2001. hepth/0009167.
[170] W. Fischler and L. Susskind. Dilaton tadpoles, string condensates and scale
invariance. Phys. Lett., B171:383, 1986.
[171] W. Fischler and L. Susskind. Dilaton tadpoles, string condensates and scale
invariance. 2. Phys. Lett., B173:262, 1986.
[172] A. Font, L. E. Ib´nez, D. L¨st, and F. Quevedo. Strong  weak coupling duality
a˜
u
and nonperturbative eﬀects in string theory. Phys. Lett., B249:35–43, 1990.
[173] A. Font, L. E. Ib´nez, H. P. Nilles, and F. Quevedo. Degenerate orbifolds. Nucl.
a˜
Phys., B307:109, 1988.
[174] A. Font, L. E. Ib´nez, H. P. Nilles, and F. Quevedo. Yukawa couplings in
a˜
degenerate orbifolds: Towards a realistic SU (3) × SU (2) × U (1) superstring.
Phys. Lett., 210B:101, 1988. [175] A. Font, L. E. Ib´nez, F. Quevedo, and A. Sierra. The construction of ’realistic’
a˜
fourdimensional strings through orbifolds. Nucl. Phys., B331:421–474, 1990.
[176] S. F¨rste. Generalized conformal invariance conditions on a sigma model of the
o
open bosonic string including the ﬁrst massive mode. Annalen Phys., 1:98, 1992.
[177] S. F¨rste. Membrany corrections to the string antistring potential in M5brane
o
theory. JHEP, 05:002, 1999. hepth/9902068.
[178] S. F¨rste and D. Ghoshal. Strings from orientifolds. Nucl. Phys., B527:95–120,
o
1998. hepth/9711039.
[179] S. F¨rste, D. Ghoshal, and S. Panda. An orientifold of the solitonic ﬁvebrane.
o
Phys. Lett., B411:46–52, 1997. hepth/9706057.
[180] S. F¨rste, D. Ghoshal, and S. Theisen. Stringy corrections to the Wilson loop
o
in N = 4 super YangMills theory. JHEP, 08:013, 1999. hepth/9903042.
[181] S. F¨rste, D. Ghoshal, and S. Theisen. Wilson loop via AdS/CFT duality. 1999.
o
hepth/0003068. BIBLIOGRAPHY 218 [182] S. F¨rste, G. Honecker, and R. Schreyer. Orientifolds with branes at angles.
o
JHEP, 06:004, 2001. hepth/0105208.
[183] S. F¨rste, G. Honecker, and R. Schreyer. Supersymmetric ZN ×ZM orientifolds in
o
4D with Dbranes at angles. Nucl. Phys., B593:127–154, 2001. hepth/0008250.
[184] S. F¨rste, A. A. Kehagias, and S. Schwager. NonAbelian Duality for Open
o
Strings. Nucl. Phys., B478:141–155, 1996. hepth/9604013.
[185] S. F¨rste, A. A. Kehagias, and S. Schwager. Non Abelian Tduality for open
o
strings. Nucl. Phys. Proc. Suppl., 56B:36–41, 1997. hepth/9610062.
[186] S. F¨rste, Z. Lalak, S. Lavignac, and H. P. Nilles. A comment on selftuning and
o
vanishing cosmological constant in the brane world. Phys. Lett., B481:360–364,
2000. hepth/0002164.
[187] S. F¨rste, Z. Lalak, S. Lavignac, and H. P. Nilles. The cosmological constant
o
problem from a braneworld perspective. JHEP, 09:034, 2000. hepth/0006139.
[188] S. F¨rste and J. Louis. Duality in string theory. Nucl. Phys. Proc. Suppl.,
o
61A:3–22, 1998. hepth/9612192.
[189] E. S. Fradkin and A. A. Tseytlin. Nonlinear electrodynamics from quantized
strings. Phys. Lett., B163:123, 1985.
[190] E. S. Fradkin and A. A. Tseytlin. Quantum string theory eﬀective action. Nucl.
Phys., B261:1–27, 1985.
[191] P. Di Francesco, P. Mathieu, and D. Senechal. Conformal ﬁeld theory. New
York, USA: Springer (1997) 890 p.
[192] D. Z. Freedman, S. S. Gubser, K. Pilch, and N. P. Warner. Renormalization
group ﬂows from holography supersymmetry and a ctheorem. Adv. Theor. Math.
Phys., 3:363–417, 1999. hepth/9904017.
[193] D. Friedan. Introduction to Polyakov’s string theory. To appear in Proc. of
Summer School of Theoretical Physics: Recent Advances in Field Theory and
Statistical Mechanics, Les Houches, France, Aug 2Sep 10, 1982.
[194] J. Fuchs and C. Schweigert. Branes: From free ﬁelds to general backgrounds.
Nucl. Phys., B530:99–136, 1998. hepth/9712257.
[195] M. Gaberdiel. Lectures on nonBPS Dirichlet branes. Class. Quant. Grav.,
17:3483–3520, 2000. hepth/0005029. BIBLIOGRAPHY 219 [196] J. P. Gauntlett. Intersecting branes. 1997. hepth/9705011.
[197] T. Gherghetta and Y. Oz. Supergravity, nonconformal ﬁeld theories and braneworlds. 2001. arXiv:hepth/0106255.
[198] T. Gherghetta and A. Pomarol. Bulk ﬁelds and supersymmetry in a slice of AdS.
Nucl. Phys., B586:141–162, 2000. arXiv:hepph/0003129.
[199] G. Gibbons, R. Kallosh, and A. Linde. Brane world sum rules. JHEP, 01:022,
2001. hepth/0011225.
[200] G. W. Gibbons and S. W. Hawking. Action integrals and partition functions in
quantum gravity. Phys. Rev., D15:2752–2756, 1977.
[201] S. B. Giddings and E. Katz. Eﬀective theories and black hole production in
warped compactiﬁcations. 2000. hepth/0009176.
[202] P. Gilkey. Invaraince theory, the heat equation and the AtiyahSinger index
theorem. CRC Press, second edition, 1995.
[203] E. G. Gimon and C. V. Johnson. K3 Orientifolds. Nucl. Phys., B477:715–745,
1996. hepth/9604129.
[204] E. G. Gimon and J. Polchinski. Consistency Conditions for Orientifolds and
DManifolds. Phys. Rev., D54:1667–1676, 1996. hepth/9601038.
[205] P. Ginsparg. Applied conformal ﬁeld theory. Lectures given at Les Houches
Summer School in Theoretical Physics, Les Houches, France, Jun 28  Aug 5,
1988.
[206] P. Ginsparg. Comment on toroidal compactiﬁcation of heterotic superstrings.
Phys. Rev., D35:648, 1987.
[207] A. Giveon. Target space duality and stringy black holes. Mod. Phys. Lett.,
A6:2843–2854, 1991.
[208] A. Giveon. On nonperturbative results in supersymmetric gauge theories: A
lecture. 1996. hepth/9611152.
[209] A. Giveon and E. Kiritsis. Axial vector duality as a gauge symmetry and topology
change in string theory. Nucl. Phys., B411:487–508, 1994. hepth/9303016.
[210] A. Giveon and D. Kutasov. Brane dynamics and gauge theory. Rev. Mod. Phys.,
71:983–1084, 1999. hepth/9802067. BIBLIOGRAPHY 220 [211] A. Giveon, M. Porrati, and E. Rabinovici. Target space duality in string theory.
Phys. Rept., 244:77–202, 1994. hepth/9401139.
[212] A. Giveon, E. Rabinovici, and G. Veneziano. Duality in string background space.
Nucl. Phys., B322:167, 1989.
[213] A. Giveon and M. Roˇek. Generalized duality in curved string backgrounds.
c
Nucl. Phys., B380:128–146, 1992. hepth/9112070.
[214] A. Giveon and M. Roˇek. On nonAbelian duality. Nucl. Phys., B421:173–190,
c
1994. hepth/9308154.
[215] W. D. Goldberger and M. B. Wise. Modulus stabilization with bulk ﬁelds. Phys.
Rev. Lett., 83:4922–4925, 1999. hepph/9907447.
[216] J. Govaerts. Quantum consistency of open string theories. Phys. Lett., B220:77,
1989.
[217] M. B. Green. Spacetime duality and Dirichlet string theory. Phys. Lett., B266:325–336, 1991.
[218] M. B. Green, J. A. Harvey, and G. Moore. Ibrane inﬂow and anomalous couplings on Dbranes. Class. Quant. Grav., 14:47–52, 1997. hepth/9605033.
[219] M. B. Green and J. H. Schwarz. Extended supergravity in tendimensions. Phys.
Lett., B122:143, 1983.
[220] M. B. Green and J. H. Schwarz. Anomaly cancellation in supersymmetric d=10
gauge theory and superstring theory. Phys. Lett., B149:117–122, 1984.
[221] M. B. Green and J. H. Schwarz. Inﬁnity cancellations in SO(32) superstring
theory. Phys. Lett., B151:21–25, 1985.
[222] M. B. Green, J. H. Schwarz, and E. Witten. Superstring theory. vol. 1: Introduction. Cambridge, Uk: Univ. Pr. ( 1987) 469 P. ( Cambridge Monographs On
Mathematical Physics).
[223] M. B. Green, J. H. Schwarz, and E. Witten. Superstring theory. vol. 2: Loop
amplitudes, anomalies and phenomenology. Cambridge, Uk: Univ. Pr. ( 1987)
596 P. ( Cambridge Monographs On Mathematical Physics).
[224] B. R. Greene. String theory on CalabiYau manifolds. 1996. hepth/9702155.
[225] B. R. Greene, A. Shapere, C. Vafa, and S. Yau. Stringy cosmic strings and
noncompact CalabiYau manifolds. Nucl. Phys., B337:1, 1990. BIBLIOGRAPHY 221 [226] J. Greensite and P. Olesen. Worldsheet ﬂuctuations and the heavy quark potential in the AdS/CFT approach. JHEP, 04:001, 1999. hepth/9901057.
[227] B. Grinstein, D. R. Nolte, and W. Skiba. Adding matter to Poincare invariant
branes. Phys. Rev., D62:086006, 2000. hepth/0005001.
[228] M. T. Grisaru, D. I. Kazakov, and D. Zanon. Five loop divergences for the N=2
supersymmetric nonlinear sigma model. Nucl. Phys., B287:189, 1987.
[229] C. Grojean, F. Quevedo, G. Tasinato, and I. Zavala C. Branes on charged
dilatonic backgrounds: Selftuning, Lorentz violations and cosmology. JHEP,
08:005, 2001. hepth/0106120.
[230] D. J. Gross and H. Ooguri. Aspects of large N gauge theory dynamics as seen
by string theory. Phys. Rev., D58:106002, 1998. hepth/9805129.
[231] S. S. Gubser. Curvature singularities: The good, the bad, and the naked. 2000.
hepth/0002160.
[232] S. S. Gubser. AdS/CFT and gravity. Phys. Rev., D63:084017, 2001. hepth/9912001.
[233] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov. Gauge theory correlators from
noncritical string theory. Phys. Lett., B428:105–114, 1998. hepth/9802109.
[234] S. S. Gubser, I. R. Klebanov, and A. A. Tseytlin. Coupling constant dependence
in the thermodynamics of N = 4 supersymmetric YangMills theory. Nucl. Phys.,
B534:202–222, 1998. hepth/9805156.
[235] M. Haack, B. K¨rs, and D. L¨st. Recent developments in string theory: From
o
u
perturbative dualities to Mtheory. 1998. hepth/9904033.
[236] M. Haack, J. Louis, and H. Singh. Massive type IIA theory on K3. JHEP,
04:040, 2001. hepth/0102110.
[237] S. Hamidi and C. Vafa. Interactions on orbifolds. Nucl. Phys., B279:465, 1987.
[238] A. Hanany and E. Witten. Type IIB superstrings, BPS monopoles, and threedimensional gauge dynamics. Nucl. Phys., B492:152–190, 1997. hepth/9611230.
[239] J. A. Harvey. Magnetic monopoles, duality, and supersymmetry. 1996. hepth/9603086.
[240] A. Hebecker. On dynamical adjustment mechanisms for the cosmological constant. 2001. hepph/0105315. BIBLIOGRAPHY 222 [241] M. Henningson and K. Skenderis. The holographic Weyl anomaly. JHEP, 07:023,
1998. hepth/9806087.
[242] S. Hewson and M. Perry. Exact nonabelian duality. Phys. Lett., B391:316–323,
1997. hepth/9603015.
[243] P. Hoˇava. Background duality of open string models. Phys. Lett., B231:251,
r
1989.
[244] P. Hoˇava. Strings on world sheet orbifolds. Nucl. Phys., B327:461, 1989.
r
[245] P. Hoˇava and E. Witten. ElevenDimensional Supergravity on a Manifold with
r
Boundary. Nucl. Phys., B475:94–114, 1996. hepth/9603142.
[246] P. Hoˇava and E. Witten. Heterotic and type I string dynamics from eleven
r
dimensions. Nucl. Phys., B460:506–524, 1996. hepth/9510209.
[247] G. T. Horowitz, I. Low, and A. Zee. Selftuning in an outgoing brane wave
model. Phys. Rev., D62:086005, 2000. hepth/0004206.
[248] G. T. Horowitz and A. Strominger. Black strings and pbranes. Nucl. Phys.,
B360:197–209, 1991.
[249] S. Hosono, A. Klemm, and S. Theisen. Lectures on mirror symmetry. 1994.
hepth/9403096.
[250] P. S. Howe and P. C. West. The complete N=2, d = 10 supergravity. Nucl.
Phys., B238:181, 1984.
[251] T. H¨bsch. CalabiYau Manifolds/A Bestiary for Physicists. World Scientiﬁc
u
(1992).
[252] C. M. Hull and P. K. Townsend. Unity of superstring dualities. Nucl. Phys.,
B438:109–137, 1995. hepth/9410167.
[253] L. E. Ib´nez, J. E. Kim, H. P. Nilles, and F. Quevedo. Orbifold compactiﬁcations
a˜
with three families of SU (3) × SU (2) × U (1)n . Phys. Lett., B191:282–286, 1987.
[254] L. E. Ib´nez, F. Marchesano, and R. Rabadan. Getting just the standard model
a˜
at intersecting branes. 2001. hepth/0105155.
[255] L. E. Ib´nez, J. Mas, H. P. Nilles, and F. Quevedo. Heterotic strings in symmetric
a˜
and asymmetric orbifold backgrounds. Nucl. Phys., B301:157, 1988. BIBLIOGRAPHY 223 [256] L. E. Ib´nez, H. P. Nilles, and F. Quevedo. Orbifolds and Wilson lines. Phys.
a˜
Lett., B187:25–32, 1987.
[257] L. E. Ib´nez, H. P. Nilles, and F. Quevedo. Reducing the rank of the gauge group
a˜
in oribifold compactiﬁcations of the heterotic string. Phys. Lett., B192:332, 1987.
[258] K. Intriligator, R. G. Leigh, and M. J. Strassler. New examples of duality in
chiral and nonchiral supersymmetric gauge theories. Nucl. Phys., B456:567–621,
1995. hepth/9506148.
[259] K. Intriligator and P. Pouliot. Exact superpotentials, quantum vacua and duality
in supersymmetric SP (Nc ) gauge theories. Phys. Lett., B353:471–476, 1995. hepth/9505006.
[260] K. Intriligator and N. Seiberg. Duality, monopoles, dyons, conﬁnement and
oblique conﬁnement in supersymmetric SO(Nc ) gauge theories. Nucl. Phys.,
B444:125–160, 1995. hepth/9503179.
[261] K. Intriligator and N. Seiberg. Lectures on supersymmetric gauge theories and
electricmagnetic duality. Nucl. Phys. Proc. Suppl., 45BC:1–28, 1996. hepth/9509066.
[262] R. A. Janik. String ﬂuctuations, AdS/CFT and the soft pomeron intercept.
Phys. Lett., B500:118–124, 2001. hepth/0010069.
[263] C. V. Johnson. Etudes on Dbranes. 1998. hepth/9812196.
[264] C. V. Johnson. Dbrane primer. 2000. hepth/0007170.
[265] B. L. Julia. Dualities in the classical supergravity limits: Dualisations, dualities
and a detour via 4k+2 dimensions. 1997. hepth/9805083.
[266] S. Kachru. Lectures on warped compactiﬁcations and stringy brane constructions. 2000. hepth/0009247.
[267] S. Kachru, M. Schulz, and E. Silverstein. Bounds on curved domain walls in 5d
gravity. Phys. Rev., D62:085003, 2000. hepth/0002121.
[268] S. Kachru, M. Schulz, and E. Silverstein. Selftuning ﬂat domain walls in 5d
gravity and string theory. Phys. Rev., D62:045021, 2000. hepth/0001206.
[269] M. Kaku. Introduction to superstrings. NEW YORK, USA: SPRINGER (1988)
568 P. (GRADUATE TEXTES IN CONTEMPORARY PHYSICS). BIBLIOGRAPHY 224 [270] M. Kaku. Introduction to superstrings and Mtheory. New York, USA: Springer
(1999) 587 p.
[271] Z. Kakushadze. ’Selftuning’ and conformality. Mod. Phys. Lett., A15:1879,
2000. hepth/0009199.
[272] Z. Kakushadze and G. Shiu. A chiral N = 1 type I vacuum in four dimensions
and its heterotic dual. Phys. Rev., D56:3686–3697, 1997. hepth/9705163.
[273] R. Kallosh. Superconformal actions in Killing gauge. 1998. hepth/9807206.
[274] R. Kallosh and A. Linde. Supersymmetry and the brane world. JHEP, 02:005,
2000. hepth/0001071.
[275] R. Kallosh and J. Rahmfeld. The GS string action on AdS5 × S 5. Phys. Lett.,
B443:143–146, 1998. hepth/9808038.
[276] R. Kallosh, J. Rahmfeld, and A. Rajaraman. Near horizon superspace. JHEP,
09:002, 1998. hepth/9805217.
[277] R. Kallosh and A. A. Tseytlin. Simplifying superstring action on AdS5 × S 5 .
JHEP, 10:016, 1998. hepth/9808088.
[278] S. Kalyana Rama. Brane world scenario with mform ﬁeld: Stabilisation of
radion modulus and self tuning solutions. Phys. Lett., B495:176–182, 2000. hepth/0010121.
[279] P. Kanti, I. I. Kogan, K. A. Olive, and M. Pospelov. Cosmological 3brane
solutions. Phys. Lett., B468:31–39, 1999. hepph/9909481.
[280] P. Kanti, K. Olive, and M. Pospelov. Static solutions for brane models with a
bulk scalar ﬁeld. Phys. Lett., B481:386–396, 2000. hepph/0002229.
[281] V. S. Kaplunovsky. Mass scales of the string uniﬁcation. Phys. Rev. Lett.,
55:1036, 1985.
[282] A. Karch. Field theory dynamics from branes in string theory. 1998. hepth/9812072.
[283] A. Karch, D. L¨st, and D. Smith. Equivalence of geometric engineering and
u
HananyWitten via fractional branes. Nucl. Phys., B533:348–372, 1998. hepth/9803232.
[284] A. Karch and L. Randall.
arXiv:hepth/0011156. Locally localized gravity. JHEP, 05:008, 2001. BIBLIOGRAPHY 225 [285] A. Karch and L. Randall. Open and closed string interpretation of SUSY CFT’s
on branes with boundaries. JHEP, 06:063, 2001. arXiv:hepth/0105132.
[286] H. Kawai, D. C. Lewellen, and S. H. H. Tye. Construction of fermionic string
models in four dimensions. Nucl. Phys., B288:1, 1987.
[287] A. Kehagias and K. Sfetsos. Deviations from the 1/r2 newton law due to extra
dimensions. Phys. Lett., B472:39–44, 2000. hepph/9905417.
[288] A. Kehagias and K. Tamvakis. A selftuning solution of the cosmological constant
problem. 2000. hepth/0011006.
[289] A. A. Kehagias. Type IIA/IIB string duality for targets with abelian isometries.
Phys. Lett., B377:241–244, 1996. hepth/9602059.
[290] K. Kikkawa and M. Yamasaki. Casimir eﬀects in superstring theories. Phys.
Lett., B149:357, 1984.
[291] H. D. Kim. A criterion for admissible singularities in brane world. Phys. Rev.,
D63:124001, 2001. hepth/0012091.
[292] J. E. Kim, B. Kyae, and H. M. Lee. A model for selftuning the cosmological
constant. Phys. Rev. Lett., 86:4223–4226, 2001. hepth/0011118.
[293] J. E. Kim, B. Kyae, and H. M. Lee. Selftuning solution of the cosmological
constant problem with antisymmetric tensor ﬁeld. 2001. hepth/0101027.
[294] Y. Kinar, E. Schreiber, J. Sonnenschein, and N. Weiss. Quantum ﬂuctuations
of Wilson loops from string models. Nucl. Phys., B583:76–104, 2000. hepth/9911123.
[295] E. Kiritsis. Introduction to superstring theory. 1997. hepth/9709062.
[296] E. Kiritsis, C. Kounnas, and D. L¨st. A large class of new gravitational and
u
axionic backgrounds for fourdimensional superstrings.
A9:1361–1394, 1994. hepth/9308124. Int. J. Mod. Phys., [297] E. Kiritsis and N. A. Obers. A New duality symmetry in string theory. Phys.
Lett., B334:67–71, 1994. hepth/9406082.
[298] E. B. Kiritsis. Duality in gauged WZW models. Mod. Phys. Lett., A6:2871–2880,
1991.
[299] I. R. Klebanov.
th/0009139. Introduction to the ads/cft correspondence. 2000. hep BIBLIOGRAPHY 226 [300] M. Klein and R. Rabadan. ZN × ZM orientifolds with and without discrete
torsion. JHEP, 10:049, 2000. hepth/0008173.
[301] M. Klein and R. Rabadan. Orientifolds with discrete torsion. JHEP, 07:040,
2000. hepth/0002103.
[302] M. Klein and R. Rabadan. D = 4, N = 1 orientifolds with vector structure.
Nucl. Phys., B596:197–230, 2001. hepth/0007087.
[303] A. Klemm, W. Lerche, and S. Theisen. Nonperturbative eﬀective actions of
N=2 supersymmetric gauge theories. Int. J. Mod. Phys., A11:1929–1974, 1996.
hepth/9505150.
[304] A. Klemm, W. Lerche, S. Yankielowicz, and S. Theisen. Simple singularities
and N=2 supersymmetric YangMills theory. Phys. Lett., B344:169–175, 1995.
hepth/9411048.
[305] C. Klimˇik and P. Severa. Dual nonAbelian duality and the Drinfeld double.
c
Phys. Lett., B351:455–462, 1995. hepth/9502122.
[306] C. Klimˇik and P. Severa. PoissonLie Tduality: Open Strings and Dbranes.
c
Phys. Lett., B376:82–89, 1996. hepth/9512124.
[307] A. Krause. A small cosmological constant and backreaction of nonﬁnetuned
parameters. 2000. hepth/0007233.
[308] A. Krause. A small cosmological constant, grand uniﬁcation and warped geometry. 2000. hepth/0006226.
[309] D. Kutasov and A. Schwimmer. On duality in supersymmetric YangMills theory. Phys. Lett., B354:315–321, 1995. hepth/9505004.
[310] D. Kutasov, A. Schwimmer, and N. Seiberg. Chiral rings, singularity theory and
electricmagnetic duality. Nucl. Phys., B459:455–496, 1996. hepth/9510222.
[311] J. M. F. Labastida and Maria A. H. Vozmediano. Bosonic strings in background
massive ﬁelds. Nucl. Phys., B312:308, 1989.
[312] J. Lauer, J. Mas, and H. P. Nilles. Duality and the role of nonperturbative eﬀects
on the world sheet. Phys. Lett., B226:251, 1989.
[313] J. Lauer, J. Mas, and H. P. Nilles. Twisted sector representations of discrete
background symmetries for twodimensional orbifolds. Nucl. Phys., B351:353–
424, 1991. BIBLIOGRAPHY 227 [314] J. Lee. Zero norm states and enlarged gauge symmetries of closed bosonic string
in background massive ﬁelds. Nucl. Phys., B336:222–244, 1990.
[315] R. G. Leigh. DiracBornInfeld action from Dirichlet sigma model. Mod. Phys.
Lett., A4:2767, 1989.
[316] R. G. Leigh and M. J. Strassler. Exactly marginal operators and duality in
fourdimensional N=1 supersymmetric gauge theory. Nucl. Phys., B447:95–136,
1995. hepth/9503121.
[317] W. Lerche. Introduction to SeibergWitten theory and its stringy origin. Nucl.
Phys. Proc. Suppl., 55B:83–117, 1997. hepth/9611190.
[318] W. Lerche, A. N. Schellekens, and N. P. Warner. Lattices and strings. Phys.
Rept., 177:1, 1989.
[319] A. Lerda and R. Russo. Stable nonBPS states in string theory: A pedagogical
review. Int. J. Mod. Phys., A15:771–820, 2000. hepth/9905006.
[320] M. Li. Dirichlet Boundary State in Linear Dilaton Background. Phys. Rev.,
D54:1644–1646, 1996. hepth/9512042.
[321] T. Li. Timelike extra dimension and cosmological constant in brane models.
Phys. Lett., B503:163–172, 2001. hepth/0009132.
[322] A. Loewy and J. Sonnenschein. On the holographic duals of N = 1 gauge dynamics. JHEP, 08:007, 2001. hepth/0103163.
[323] J. C. Long, H. W. Chan, and John C. Price. Experimental status of gravitationalstrength forces in the subcentimeter regime. Nucl. Phys., B539:23–34, 1999.
hepph/9805217.
[324] J. Louis. Recent developments in superstring phenomenology. 1992. hep ph/9205226.
[325] J. Louis. Phenomenological aspects of string theory. 1998. Published in Trieste
1998, Nonperturbative aspects of strings, branes and supersymmetry, 178208.
[326] C. Lovelace. Strings in curved space. Phys. Lett., B135:75, 1984.
[327] C. Lovelace. Stability of string vacua. 1. a new picture of the renormalization
group. Nucl. Phys., B273:413, 1986.
[328] H. Luckock. Quantum geometry of strings with boundaries. Ann. Phys., 194:113,
1989. BIBLIOGRAPHY 228 [329] A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram. Heterotic Mtheory in
ﬁve dimensions. Nucl. Phys., B552:246–290, 1999. hepth/9806051.
[330] A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram. The universe as a domain
wall. Phys. Rev., D59:086001, 1999. hepth/9803235.
[331] D. L¨st and S. Theisen. Lectures on string theory. Berlin, Germany: Springer
u
(1989) 346 p. (Lecture notes in physics, 346).
[332] J. Lykken and L. Randall. The shape of gravity. JHEP, 06:014, 2000. hepth/9908076.
[333] J. D. Lykken. Introduction to supersymmetry. 1996. hepth/9612114.
[334] R. Maartens. Geometry and dynamics of the braneworld. 2001. arXiv:grqc/0101059.
[335] J. Maldacena. The large N limit of superconformal ﬁeld theories and supergravity. Adv. Theor. Math. Phys., 2:231–252, 1998. hepth/9711200.
[336] J. Maldacena. Wilson loops in large N ﬁeld theories. Phys. Rev. Lett., 80:4859–
4862, 1998. hepth/9803002.
[337] J. M. Maldacena. Black holes in string theory. 1996. hepth/9607235.
[338] J. M. Maldacena. Black holes and Dbranes. Nucl. Phys. Proc. Suppl., 61A:111–
123, 1998. hepth/9705078.
[339] K. A. Meissner and G. Veneziano. Manifestly O(d,d) invariant approach to
spacetime dependent string vacua. Mod. Phys. Lett., A6:3397–3404, 1991. hepth/9110004.
[340] R. R. Metsaev and A. A. Tseytlin. Type IIB superstring action in AdS5 × S 5
background. Nucl. Phys., B533:109–126, 1998. hepth/9805028. [341] K. A. Milton. Constraints on extra dimensions from cosmological and terrestrial
measurements. Grav. Cosmol., 6:1–10, 2000. hepth/0107241.
[342] K. A. Milton, R. Kantowski, C. Kao, and Y. Wang. Constraints on extra dimensions from cosmological and terrestrial measurements. 2001. hepph/0105250.
[343] N. Mohammedi. On NonAbelian Duality in Sigma Models. B375:149–153, 1996. hepth/9512126. Phys. Lett., BIBLIOGRAPHY 229 [344] T. Mohaupt. Black holes in supergravity and string theory. Class. Quant. Grav.,
17:3429–3482, 2000. hepth/0004098.
[345] T. Mohaupt. Black hole entropy, special geometry and strings. Fortsch. Phys.,
49:3–161, 2001. hepth/0007195.
[346] C. Montonen and D. Olive. Magnetic monopoles as gauge particles? Phys. Lett.,
B72:117, 1977.
[347] W. M¨ck. Ideas on the semiclassical path integral over embedded manifolds.
u
Fortsch. Phys., 49:607–615, 2001.
[348] R. C. Myers. Black holes and string theory. 2001. grqc/0107034.
[349] W. Nahm. Supersymmetries and their representations. Nucl. Phys., B135:149,
1978.
[350] W. Nahm and K. Wendland. A hiker’s guide to K3: Aspects of n = (4,4)
superconformal ﬁeld theory with central charge c = 6. Commun. Math. Phys.,
216:85–138, 2001. hepth/9912067.
[351] S. Naik. Improved heavy quark potential at ﬁnite temperature from antide
Sitter supergravity. Phys. Lett., B464:73–76, 1999. hepth/9904147.
[352] K. S. Narain. New heterotic string theories in uncompactiﬁed dimensions < 10.
Phys. Lett., B169:41, 1986.
[353] R. I. Nepomechie. Magnetic monopoles from antisymmetric tensor gauge ﬁelds.
Phys. Rev., D31:1921, 1985.
[354] H. P. Nilles. Strings on orbifolds: An introduction. Lectures given at Int. Summer
School on Conformal Invariance and String Theory, Poiana Brasov, Romania,
Sep 112, 1987.
[355] H. P. Nilles. Supersymmetry, supergravity and particle physics. Phys. Rept.,
110:1, 1984.
[356] N. A. Obers and B. Pioline. Uduality and Mtheory. Phys. Rept., 318:113–225,
1999. arXiv:hepth/9809039.
[357] D. O’Driscoll. General abelian orientifold models and one loop amplitudes. 1998.
hepth/9801114.
[358] D. I. Olive. Exact electromagnetic duality. Nucl. Phys. Proc. Suppl., 45A:88–102,
1996. hepth/9508089. BIBLIOGRAPHY 230 [359] H. Ooguri and Z. Yin. TASI lectures on perturbative string theories. 1996.
hepth/9612254.
[360] H. Osborn. Topological charges for N=4 supersymmetric gauge theories and
monopoles of spin 1. Phys. Lett., B83:321, 1979.
[361] J. Pawelczyk and S. Theisen. AdS5 × S 5 black hole metric at O α3 . JHEP,
09:010, 1998. hepth/9808126.
[362] A. W. Peet. The Bekenstein formula and string theory (Nbrane theory). Class.
Quant. Grav., 15:3291–3338, 1998. hepth/9712253.
[363] M. PerezVictoria. RandallSundrum models and the regularized AdS/CFT correspondence. JHEP, 05:064, 2001. arXiv:hepth/0105048.
[364] I. Pesando. A kappa gauge ﬁxed type IIB superstring action on AdS5 × S 5 .
JHEP, 11:002, 1998. hepth/9808020.
[365] M. E. Peskin.
th/9702094. Duality in supersymmetric YangMills theory. 1997. hep [366] J. L. Petersen. Introduction to the Maldacena conjecture on AdS/CFT. Int. J.
Mod. Phys., A14:3597–3672, 1999. hepth/9902131.
[367] J. Plefka and M. Staudacher. Two loops to two loops in N = 4 supersymmetric
YangMills theory. 2001. hepth/0108182.
[368] J. Polchinski. What is string theory? 1994. hepth/9411028.
[369] J. Polchinski. DirichletBranes and RamondRamond Charges. Phys. Rev. Lett.,
75:4724–4727, 1995. hepth/9510017.
[370] J. Polchinski. TASI lectures on Dbranes. 1996. hepth/9611050.
[371] J. Polchinski. String Theory, volume 1 & 2. Cambridge University Press, 1998.
[372] J. Polchinski and Y. Cai. Consistency of open superstring theories. Nucl. Phys.,
B296:91, 1988.
[373] J. Polchinski, S. Chaudhuri, and C. V. Johnson. Notes on Dbranes. 1996.
hepth/9602052.
[374] J. Polchinski and E. Witten. Evidence for Heterotic  Type I String Duality.
Nucl. Phys., B460:525–540, 1996. hepth/9510169. BIBLIOGRAPHY 231 [375] A. M. Polyakov. Quantum geometry of bosonic strings. Phys. Lett., B103:207–
210, 1981.
[376] A. M. Polyakov. Quantum geometry of fermionic strings. Phys. Lett., B103:211–
213, 1981.
[377] G. Pradisi. Type I vacua from diagonal Z3 orbifolds. Nucl. Phys., B575:134–150,
2000. hepth/9912218.
[378] G. Pradisi and A. Sagnotti. Open string orbifolds. Phys. Lett., B216:59, 1989.
[379] M. K. Prasad and C. M. Sommerﬁeld. An exact classical solution for the ’t Hooft
monopole and the JuliaZee dyon. Phys. Rev. Lett., 35:760–762, 1975.
[380] F. Quevedo. Superstrings and physics? Lectures given at Workshop on Gauge
Theories, Applied Supersymmetry and Quantum Gravity, London, England, 510 Jul 1996.
[381] F. Quevedo. Superstring phenomenology: An overview. Nucl. Phys. Proc. Suppl.,
62:134–143, 1998. hepph/9707434.
[382] R. Rabadan. Branes at angles, torons, stability and supersymmetry. 2001. hepth/0107036.
[383] R. Rabadan and A. M. Uranga. Type IIB orientifolds without untwisted tadpoles, and nonBPS Dbranes. JHEP, 01:029, 2001. hepth/0009135.
[384] A. Rajaraman and M. Rozali. On the quantization of the GS string on AdS5 × S 5 .
Phys. Lett., B468:58–64, 1999. hepth/9902046.
[385] L. J. Randall and R. Sundrum. An alternative to compactiﬁcation. Phys. Rev.
Lett., 83:4690–4693, 1999. hepth/9906064.
[386] L. J. Randall and R. Sundrum. A large mass hierarchy from a small extra
dimension. Phys. Rev. Lett., 83:3370–3373, 1999. hepph/9905221.
[387] R. Rattazzi and A. Zaﬀaroni. Comments on the holographic picture of the
RandallSundrum model. JHEP, 04:021, 2001. arXiv:hepth/0012248.
[388] A. Recknagel and V. Schomerus. Dbranes in Gepner models. Nucl. Phys.,
B531:185–225, 1998. hepth/9712186.
[389] S. Rey. The conﬁning phase of superstrings and axionic strings. Phys. Rev.,
D43:526–538, 1991. BIBLIOGRAPHY 232 [390] S. Rey, S. Theisen, and J. Yee. WilsonPolyakov loop at ﬁnite temperature in
large N gauge theory and antide Sitter supergravity. Nucl. Phys., B527:171–186,
1998. hepth/9803135.
[391] S. Rey and J. Yee. Macroscopic strings as heavy quarks in large N gauge theory
and antide Sitter supergravity. 1998. hepth/9803001.
[392] M. Roˇek and E. Verlinde.
c Duality, quotients, and currents. Nucl. Phys., B373:630–646, 1992. hepth/9110053.
[393] V. A. Rubakov. Large and inﬁnite extra dimensions: An Introduction. 2001.
hepph/0104152.
[394] V. A. Rubakov and M. E. Shaposhnikov. Do we live inside a domain wall? Phys.
Lett., B125:136–138, 1983.
[395] V. A. Rubakov and M. E. Shaposhnikov. Extra spacetime dimensions: Towards
a solution to the cosmological constant problem. Phys. Lett., B125:139, 1983.
[396] A. Sagnotti. Open strings and their symmetry groups. Talk presented at
the Cargese Summer Institute on Non Perturbative Methods in Field Theory,
Cargese, France, Jul 1630, 1987.
[397] A. Sagnotti. Surprises in openstring perturbation theory. Nucl. Phys. Proc.
Suppl., 56B:332–343, 1997. hepth/9702093.
[398] N. Sakai and I. Senda. Vacuum energies of string compactiﬁed on torus. Prog.
Theor. Phys., 75:692, 1986.
[399] A. Salam and E. Sezgin. Supergravities in diverse dimensions. vol. 1, 2. Amsterdam, Netherlands: NorthHolland (1989) 1499 p. Singapore, Singapore: World
Scientiﬁc (1989) 1499 p.
[400] C. Schmidhuber. Dbrane actions. Nucl. Phys., B467:146–158, 1996. hepth/9601003.
[401] C. Schmidhuber. AdS5 and the 4d cosmological constant. Nucl. Phys., B580:140–
146, 2000. hepth/9912156.
[402] C. Schmidhuber. Micrometer gravitinos and the cosmological constant. Nucl.
Phys., B585:385–394, 2000. hepth/0005248.
[403] C. Schmidhuber. Brane supersymmetry breaking and the cosmological constant:
Open problems. 2001. hepth/0104131. BIBLIOGRAPHY 233 [404] V. Schomerus. Dbranes and deformation quantization. JHEP, 06:030, 1999.
hepth/9903205.
[405] J. H. Schwarz. Covariant ﬁeld equations of chiral N=2 D = 10 supergravity.
Nucl. Phys., B226:269, 1983.
[406] J. H. Schwarz. An SL(2, Z) multiplet of type IIB superstrings. Phys. Lett.,
B360:13–18, 1995. hepth/9508143.
[407] J. H. Schwarz. Lectures on superstring and M theory dualities. Nucl. Phys.
Proc. Suppl., 55B:1–32, 1997. hepth/9607201.
[408] J. H. Schwarz. The status of string theory. 1997. hepth/9711029.
[409] J. H. Schwarz and A. Sen. Duality symmetries of 4D heterotic strings. Phys.
Lett., B312:105–114, 1993. hepth/9305185.
[410] J. H. Schwarz and A. Sen. Duality symmetric actions. Nucl. Phys., B411:35–63,
1994. hepth/9304154.
[411] J. H. Schwarz and P. C. West. Symmetries and transformations of chiral N=2
D = 10 supergravity. Phys. Lett., B126:301, 1983.
[412] N. Seiberg. Exact results on the space of vacua of fourdimensional SUSY gauge
theories. Phys. Rev., D49:6857–6863, 1994. hepth/9402044.
[413] N. Seiberg. Electric  magnetic duality in supersymmetric nonAbelian gauge
theories. Nucl. Phys., B435:129–146, 1995. hepth/9411149.
[414] N. Seiberg. The power of duality: Exact results in 4d SUSY ﬁeld theory. 1995.
hepth/9506077.
[415] N. Seiberg and E. Witten. Spin structures in string theory. Nucl. Phys., B276:272, 1986.
[416] N. Seiberg and E. Witten. Electric  magnetic duality, monopole condensation, and conﬁnement in N=2 supersymmetric YangMills theory. Nucl. Phys.,
B426:19–52, 1994. hepth/9407087.
[417] N. Seiberg and E. Witten. Monopoles, duality and chiral symmetry breaking in
N=2 supersymmetric QCD. Nucl. Phys., B431:484–550, 1994. hepth/9408099.
[418] N. Seiberg and E. Witten. String theory and noncommutative geometry. JHEP,
09:032, 1999. hepth/9908142. BIBLIOGRAPHY 234 [419] A. Sen. Equations of motion for the heterotic string theory from the conformal
invariance of the sigma model. Phys. Rev. Lett., 55:1846, 1985.
[420] A. Sen. Electric magnetic duality in string theory. Nucl. Phys., B404:109–126,
1993. hepth/9207053.
[421] A. Sen. Dyon  monopole bound states, selfdual harmonic forms on the multi monopole moduli space, and SL(2, Z) invariance in string theory. Phys. Lett.,
B329:217–221, 1994. hepth/9402032.
[422] A. Sen. Strong  weak coupling duality in fourdimensional string theory. Int.
J. Mod. Phys., A9:3707–3750, 1994. hepth/9402002.
[423] A. Sen. An introduction to nonperturbative string theory. 1998. hep th/9802051.
[424] A. Sen. NonBPS Dbranes in string theory. Class. Quant. Grav., 17:1251–1256,
2000.
[425] A. Shapere and F. Wilczek. Selfdual models with theta terms. Nucl. Phys.,
B320:669, 1989.
[426] M. Shifman. Nonperturbative dynamics in supersymmetric gauge theories. Prog.
Part. Nucl. Phys., 39:1–116, 1997. hepth/9704114.
[427] T. Shiromizu, T. Torii, and D. Ida.
arXiv:hepth/0105256. Braneworld and holography. 2001. [428] K. Skenderis. Black holes and branes in string theory. Lect. Notes Phys., 541:325–
364, 2000. hepth/9901050.
[429] M. F. Sohnius. Introducing supersymmetry. Phys. Rept., 128:39–204, 1985.
[430] J. Sonnenschein. What does the string / gauge correspondence teach us about
Wilson loops? 1999. hepth/0003032.
[431] K. S. Stelle. Lectures on supergravity pbranes. 1996. hepth/9701088.
[432] H. Stephani and (Ed. ) J. Steward. General relativity. An introduction to the
theory of the gravitational ﬁeld. Cambridge, Uk: Univ. Pr. ( 1982) 298 P. (
Transl. From German By M. Pollock and J. Stewart).
[433] A. Strominger. Heterotic solitons. Nucl. Phys., B343:167–184, 1990. BIBLIOGRAPHY 235 [434] L. Susskind and E. Witten. The holographic bound in antide Sitter space. 1998.
hepth/9805114.
[435] G. ’t Hooft. A planar diagram theory for strong interactions. Nucl. Phys.,
B72:461, 1974.
[436] R. Tatar. Dualities in 4D theories with product gauge groups from brane conﬁgurations. Phys. Lett., B419:99–106, 1998. hepth/9704198.
[437] IV Taylor, W. Lectures on Dbranes, gauge theory and M(atrices). 1997. hepth/9801182.
[438] C. Teitelboim. Monopoles of higher rank. Phys. Lett., B167:69, 1986.
[439] L. Thorlacius. Introduction to Dbranes. Nucl. Phys. Proc. Suppl., 61A:86–98,
1998. hepth/9708078.
[440] P. K. Townsend. The elevendimensional supermembrane revisited. Phys. Lett.,
B350:184–187, 1995. hepth/9501068.
[441] P. K. Townsend. Four lectures on Mtheory. 1996. hepth/9612121.
[442] P. K. Townsend. Black holes. 1997. grqc/9707012.
[443] P. K. Townsend. Mtheory from its superalgebra. 1997. hepth/9712004.
[444] P. K. Townsend. Brane theory solitons. 1999. hepth/0004039.
[445] A. A. Tseytlin. Sigma models and renormalization of string loops. Lectures given
at 1989 Trieste Spring School on Superstrings, Trieste, Italy, Apr 314, 1989.
[446] A. A. Tseytlin. Sigma model Weyl invariance conditions and string equations of
motion. Nucl. Phys., B294:383, 1987.
[447] A. A. Tseytlin. On the renormalization group approach to string equations of
motion. Int. J. Mod. Phys., A4:4249, 1989.
[448] A. A. Tseytlin. Duality and dilaton. Mod. Phys. Lett., A6:1721–1732, 1991.
[449] A. A. Tseytlin. BornInfeld action, supersymmetry and string theory. 1999.
hepth/9908105.
[450] E. Tyurin. NonAbelian axial  vector duality: A Geometric description. Phys.
Lett., B364:157–162, 1995. BIBLIOGRAPHY 236 [451] C. Vafa. Modular invariance and discrete torsion on orbifolds. Nucl. Phys.,
B273:592, 1986.
[452] C. Vafa. Lectures on strings and dualities. 1997. hepth/9702201.
[453] E. Verlinde and H. Verlinde. RGﬂow, gravity and the cosmological constant.
JHEP, 05:034, 2000. hepth/9912018.
[454] H. Verlinde. Holography and compactiﬁcation. Nucl. Phys., B580:264–274, 2000.
hepth/9906182.
[455] S. Weinberg. The cosmological constant problem. Rev. Mod. Phys., 61:1–23,
1989.
[456] P. West. Introduction to supersymmetry and supergravity. Singapore, Singapore:
World Scientiﬁc (1990) 425 p.
[457] E. Witten. Dyons of charge e theta / 2 pi. Phys. Lett., B86:283–287, 1979.
[458] E. Witten. String theory dynamics in various dimensions. Nucl. Phys., B443:85–
126, 1995. hepth/9503124.
[459] E. Witten. Solutions of fourdimensional ﬁeld theories via Mtheory. Nucl. Phys.,
B500:3–42, 1997. hepth/9703166.
[460] E. Witten. Antide Sitter space and holography. Adv. Theor. Math. Phys.,
2:253–291, 1998. hepth/9802150.
[461] E. Witten. Antide Sitter space, thermal phase transition, and conﬁnement in
gauge theories. Adv. Theor. Math. Phys., 2:505–532, 1998. hepth/9803131.
[462] E. Witten. The cosmological constant from the viewpoint of string theory. 2000.
hepph/0002297.
[463] E. Witten and D. Olive. Supersymmetry algebras that include topological
charges. Phys. Lett., B78:97, 1978.
[464] D. Youm. Black holes and solitons in string theory. Phys. Rept., 316:1–232,
1999. hepth/9710046.
[465] D. Youm. Bulk ﬁelds in dilatonic and selftuning ﬂat domain walls. Nucl. Phys.,
B589:315–336, 2000. hepth/0002147.
[466] K. Zarembo. Wilson loop correlator in the AdS/CFT correspondence. Phys.
Lett., B459:527–534, 1999. hepth/9904149. BIBLIOGRAPHY 237 [467] K. Zarembo. String breaking from ladder diagrams in SYM theory. JHEP,
03:042, 2001. hepth/0103058.
[468] M. Zucker. Supersymmetric brane world scenarios from oﬀshell supergravity.
Phys. Rev., D64:024024, 2001. hepth/0009083.
[469] G. Zwart. Fourdimensional N = 1 ZN × ZM orientifolds. Nucl. Phys., B526:378–
392, 1998. hepth/9708040. ...
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