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arXiv:hep-th/0110055 v3 3 Jan 2002 Strings, Branes and Extra Dimensions Stefan F¨orste Physikalisches Institut, Universit¨at Bonn Nussallee 12, D-53115 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, D-branes 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 4 2.1 The Fundamental String . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Worldsheet Actions . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1.1 The closed bosonic string . . . . . . . . . . . . . . . . 4 2.1.1.2 Worldsheet supersymmetry . . . . . . . . . . . . . . . 7 2.1.1.3 Space-time supersymmetric string . . . . . . . . . . . 10 2.1.2 Quantization of the fundamental string . . . . . . . . . . . . . 13 2.1.2.1 The closed bosonic string . . . . . . . . . . . . . . . . 14 2.1.2.2 Type II strings . . . . . . . . . . . . . . . . . . . . . . 19 2.1.2.3 The heterotic string . . . . . . . . . . . . . . . . . . . 25 2.1.3 Strings in non-trivial backgrounds . . . . . . . . . . . . . . . . 28 2.1.4 Perturbative expansion and effective actions . . . . . . . . . . . 36 2.1.5 Toroidal Compactification and T-duality . . . . . . . . . . . . . 42 2.1.5.1 Kaluza-Klein compactification of a scalar field . . . . 42 2.1.5.2 The bosonic string on a circle . . . . . . . . . . . . . . 43 2.1.5.3 T-duality in non trivial backgrounds . . . . . . . . . . 46 2.1.5.4 T-duality for superstrings . . . . . . . . . . . . . . . . 48 2.2 Orbifold fixed planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.1 The bosonic string on an orbicircle . . . . . . . . . . . . . . . . 51 2.2.2 Type IIB on T 4 /Z 2 . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.3 Comparison with type IIB on K3 . . . . . . . . . . . . . . . . . 57 2.3 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.1 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . 61 2.3.1.2 Quantization of the open string ending on a single D- brane . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.1.3 Number of ND directions and GSO projection . . . . 67 2.3.1.4 Multiple parallel D-branes – Chan Paton factors . . . 68 2.3.2 D-brane interactions . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3.3 D-brane actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.3.3.1 Open strings in non-trivial backgrounds . . . . . . . . 79 2.3.3.2 Toroidal compactification and T-duality for open strings 85 2.3.3.3 RR fields . . . . . . . . . . . . . . . . . . . . . . . . . 91 i CONTENTS ii 2.3.3.4 Noncommutative geometry . . . . . . . . . . . . . . . 93 2.4 Orientifold fixed planes . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.4.1 Unoriented closed strings . . . . . . . . . . . . . . . . . . . . . 98 2.4.2 O-plane interactions . . . . . . . . . . . . . . . . . . . . . . . . 101 2.4.2.1 O-plane/O-plane interaction, or the Klein bottle . . . 101 2.4.2.2 D-brane/O-plane interaction, or the M¨obius strip . . 107 2.4.3 Compactifying the transverse dimensions . . . . . . . . . . . . 112 2.4.3.1 Type I/type I strings . . . . . . . . . . . . . . . . . . 113 2.4.3.2 Orbifold compactification . . . . . . . . . . . . . . . . 116 3 Non-Perturbative description of branes 124 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.2 Universal Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.2.1 The fundamental string . . . . . . . . . . . . . . . . . . . . . . 127 3.2.2 The NS five brane . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.3 Type II branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4 Applications 138 4.1 String dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.2 Dualities in Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.3 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.3.1 The conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.3.2 Wilson loop computation . . . . . . . . . . . . . . . . . . . . . 149 4.3.2.1 Classical approximation . . . . . . . . . . . . . . . . . 149 4.3.2.2 Stringy corrections . . . . . . . . . . . . . . . . . . . . 154 4.4 Strings at a TeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.4.1 Corrections to Newton’s law . . . . . . . . . . . . . . . . . . . . 165 5 Brane world setups 167 5.1 The Randall Sundrum models . . . . . . . . . . . . . . . . . . . . . . . 167 5.1.1 The RS1 model with two branes . . . . . . . . . . . . . . . . . 167 5.1.1.1 A proposal for radion stabilization . . . . . . . . . . . 171 5.1.2 The RS2 model with one brane . . . . . . . . . . . . . . . . . . 174 5.1.2.1 Corrections to Newton’s law . . . . . . . . . . . . . . 175 5.1.2.2 and the holographic principle . . . . . . . . . . . . 179 5.1.2.3 The RS2 model with two branes . . . . . . . . . . . . 181 5.2 Inclusion of a bulk scalar . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2.1 A solution generating technique . . . . . . . . . . . . . . . . . . 183 5.2.2 Consistency conditions . . . . . . . . . . . . . . . . . . . . . . . 186 5.2.3 The cosmological constant problem . . . . . . . . . . . . . . . . 188 5.2.3.1 An example . . . . . . . . . . . . . . . . . . . . . . . . 189 5.2.3.2 A no go theorem . . . . . . . . . . . . . . . . . . . . . 193 CONTENTS iii 6 Bibliography and further reading 197 6.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.1.1 Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 6.1.2 Review articles . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.1.3 Research papers . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.2.1 Review articles . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.2.2 Research Papers . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.3.1 Review articles . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.3.2 Research papers . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.4.1 Review articles . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.4.2 Research papers . . . . . . . . . . . . . . . . . . . . . . . . . . 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 interac- tions 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 mem- brane 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 fea- tures 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 di- mensional supersymmetric extensions of) the Einstein Hilbert theory. The concept 1 1. Introduction 2 of compactifying extra dimensions is introduced and special stringy features are em- phasized. 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 D-branes, branes on which open string excitations live. We compute the tensions and charges of D-branes and derive an effective field theory on the world volume of the D-brane. 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 D-branes are candidates for phenomenologically in- teresting 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 D-branes. 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 M-theory. 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 D-branes. 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 D-branes 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 The Fundamental String 2.1.1 Worldsheet Actions 2.1.1.1 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 d 2 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, g αβ = G µν (X) ∂ α X µ ∂ β X ν , (2.1.1.2) 4 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 con- struction, the action (2.1.1.1) is invariant under reparametrizations of the worldsheet. Alternatively, we could have introduced an independent metric γ αβ on the world- sheet. This enables us to write the action (2.1.1.1) in an equivalent form, S = − 1 4πα d 2 σ √ −γγ αβ G µν ∂ α X µ ∂ β X ν . (2.1.1.3) For the target space metric we will mostly use the Minkowski metric η µν in the present chapter. Varying (2.1.1.3) with respect to γ αβ yields the energy momentum tensor, T αβ = − 4πα √ −γ δS δγ αβ = ∂ α X µ ∂ β X µ − 1 2 γ αβ γ δγ ∂ δ X µ ∂ γ X µ , (2.1.1.4) where the target space index µ is raised and lowered with G µν = η µν . Thus, the γ αβ equation of motion, T αβ = 0, equates γ αβ 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 G µν = η µν 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 G µν . In our case G µν = η µν this is nothing but d dimensional Poincar´e invariance. From 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 γ αβ → e ϕ(τ,σ) γ αβ . (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, γ αβ = η αβ . (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 version 1 of (2.1.1.3) is S = 1 2πα dσ + dσ − ∂ − X µ ∂ + X µ . (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 +− = 0 2 . However, the other γ αβ 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) are 3 ∂ + ∂ − 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 2 ˜σ + σ + + ˜σ − σ − , (2.1.1.12) which is the general solution to (2.1.1.11)). Hence, without loss of generality we can fix X + = 1 √ 2 X 0 + X 1 = x + + p + τ, (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 X − = 1 √ 2 X 0 − X 1 (2.1.1.14) as a function of X i (i = 2, . . . , d − 1) uniquely up to an integration constant corre- sponding to the center of mass position in the minus direction. Thus we are left with 1 Gauge fixing means imposing (2.1.1.6). 2 The corresponding symmetry is called conformal symmetry. It means that the action is invari- ant 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 D-branes in section 2.3. [...]... 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 non-trivial backgrounds 2.1.3 28 Strings in non-trivial backgrounds In the previous sections we... 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 (Gliozzi-Scherk-Olive) projection To specify what this projection does in the NS sector ˜ we introduce fermion number operators F (F ) counting... 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 bi , and an analogous expression for F −r r The worldsheet has the... 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. .. 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 Yang-Mills field The R sector state fills in the fermions needed for N = 1 supersymmetry in ten dimensions This corresponds to the other known N... 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... 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,... (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... 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... 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 . arXiv:hep-th/0110055 v3 3 Jan 2002 Strings, Branes and Extra Dimensions Stefan F¨orste Physikalisches Institut, Universit¨at Bonn Nussallee 12, D-53115 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