DBI Analysis of Open String Bound States on Non-compact D-branes

Master Thesis<strong>DBI</strong> <strong>Analysis</strong> <strong>of</strong> <strong>Open</strong> <strong>String</strong> <strong>Bound</strong> <strong>States</strong> onNon-compact D-branesVrije Universiteit BrusselFaculty <strong>of</strong> Science & Bio-engineering SciencesDepartment <strong>of</strong> PhysicsAuthor:Laurent MertensAdvisors:Pr<strong>of</strong>. Dr. Alexander SevrinDr. Raphael Benichou2009-2010

⌈There’s a fragile tensionThat’s keeping us goingIt may not last foreverBut oh when it’s flowingThere’s something magical in the airSomething so tragic we have to careThere’s a strange obsessionThat’s drawing us nearerWe don’t understand itIt never gets clearerThere’s something mystical in our genesSo simplistic it kicks and screamsOh when we’re teeteringOn the edge <strong>of</strong> collapseNothing can keep us downThere’s a dizzying feelingThat’s keeping us flyingThrough glittering galasWithout even tryingThere’s something radical in our handsNothing logical to our plans⌋[ Depeche Mode (M. Gore) - Fragile Tension ]

ContentsAcknowledgements 1Samenvatting 2I General Introduction 31 Introduction 41.1 <strong>String</strong> theory in a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 A brief history <strong>of</strong> string theory . . . . . . . . . . . . . . . . . . . . . . . 61.3 Guideline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8II Theoretical Introduction 112 Bosonic <strong>String</strong>s 122.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 <strong>Bound</strong>ary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 <strong>Open</strong> strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Solutions and expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.1 Closed string expansion . . . . . . . . . . . . . . . . . . . . . . . 192.5.2 <strong>Open</strong> string expansion . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.1 Covariant approach . . . . . . . . . . . . . . . . . . . . . . . . . 212.6.2 Energy-momentum tensor and Hamiltonian . . . . . . . . . . . . 232.6.3 Virasoro algebra and physical states . . . . . . . . . . . . . . . . 272.6.4 Light-cone gauge quantization and spectrum . . . . . . . . . . . 292.7 Oriented vs. unoriented strings . . . . . . . . . . . . . . . . . . . . . . . 362.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37i

3 Superstrings 383.1 Superstring action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3 <strong>Bound</strong>ary conditions and expansions . . . . . . . . . . . . . . . . . . . . 413.3.1 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.2 <strong>Open</strong> strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 Quantization and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.1 <strong>Open</strong> strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4.2 GSO projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.3 Closed strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Conformal Invariance 534.1 Conformal Field Theory Jr. . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 <strong>String</strong> Theory vs. QFT vs. General Relativity . . . . . . . . . . . . . . . 564.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Branes 605.1 Dp-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1.1 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1.2 D-branes in Type IIA and Type IIB theories . . . . . . . . . . . 685.1.3 T-duality in Type IIA and Type IIB theories . . . . . . . . . . . 705.1.4 Dirac-Born-Infeld Action . . . . . . . . . . . . . . . . . . . . . . 705.2 NS5-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73III Research 756 Setting 766.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.2 What’s in a title? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2.1 Compact vs. non-compact D-branes . . . . . . . . . . . . . . . . 776.2.2 <strong>Bound</strong> states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2.3 <strong>DBI</strong> approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Geometrical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.4 Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 How to recognize bound states . . . . . . . . . . . . . . . . . . . . . . . 837 A First Example 857.1 D1-brane on the cigar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1.1 Metric and action . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1.2 Classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.1.3 Adding perturbations . . . . . . . . . . . . . . . . . . . . . . . . 87ii

7.1.4 Perturbed equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . 897.1.5 Solutions <strong>of</strong> the PDE . . . . . . . . . . . . . . . . . . . . . . . . . 897.2 D1-brane on the trumpet . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2.1 Metric and action . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.2 Classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.3 Adding perturbations . . . . . . . . . . . . . . . . . . . . . . . . 937.2.4 Perturbed equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . 947.2.5 Solutions <strong>of</strong> the PDE . . . . . . . . . . . . . . . . . . . . . . . . . 948 Computations 968.1 Metric and field strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 968.2 Classical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.3 Field strength variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.4 Regularized Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 998.5 Equations <strong>of</strong> motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.6 Road to differential equation . . . . . . . . . . . . . . . . . . . . . . . . 1018.7 Differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028.8 Hypergeometric equation? . . . . . . . . . . . . . . . . . . . . . . . . . . 1039 Conclusion & Final Thoughts 105Bibliography 107List <strong>of</strong> Abbreviations 109iii

AcknowledgementsI would hereby like to express my deepest and earnest gratitude towards my advisor,Pr<strong>of</strong>. Dr. Alexander Sevrin, for the leap <strong>of</strong> faith he took in accepting me as his thesisstudent so late on, and on such short notice. I can only hope that after having readthrough this document, he does not bury his head in his hands, shaking it in questioningwhat could possibly have overcome him in taking that decision.Sincerest thanks also to my co-advisor, Dr. Raphaël Benichou, for his truely endlesspatience and countless precious time spent answering questions that should have gottenme sent back to my freshman year (if not even further back!), not slapping me when Icautiously explored the extend <strong>of</strong> his tolerance, steady guidance over the past monthsand valuable advice given during these, specifically with regard to the writing <strong>of</strong> thisthesis.Last but not least, I am indebted to Angelos Fotopoulos for communicated informationwith regards to the obtainment <strong>of</strong> the results reported in [6].1

SamenvattingDe zogehete snaartheorie is een vooralsnog puur theoretisch raamwerk dat als ambitieheeft de vier fundamentele krachten van de Natuur te unificeren. Het stuit daarbij echterop enige problemen, één daarvan zijnde dat de formulering die aanleiding geeft tot defysica zoals beschreven door het Standaard Model zichzelf nog niet heeft prijsgegeven.Een eerste stap naar het vinden van die formulering is het terugvinden van alle elementenaanwezig in het Standaard Model. Eén van die elementen zijn massieve ijkdeeltjes. Eenander probleem is dat snaartheorie als biotoop een tien-dimensionale ruimte-tijd heeft,en dat de vertaling van de fysica in deze tien-dimensionale ruimte-tijd naar een vierdimensionaleruimte-tijd ook nog steeds niet volledig gekend is.In [3] werd o.a. een welbepaalde tien-dimensionale configuratie onder de loep genomendie resulteert in een massief ijkdeeltje dat leeft in een vier-dimensionale Minkowskiruimte. Dit massief ijkdeeltje manifesteert zichzelf als een aangeslagen toestand van eenD4-braan. De berekeningen die dit resultaat tot gevolg hadden werden uitgevoerd in eenbi-dimensionaal conform veldentheoretisch raamwerk. Dit raamwerk heeft als voordeelexact te zijn, maar heeft als nadeel dat het slechts voor een beperkt aantal configuratiesgekend is. Anderzijds bestaat er een laag energetische braan actie, de Dirac-Born-Infeldactie, die alhoewel slechts nauwkeurig bij lage snaarkoppeling, wel éénvoudig te veralgemenenvalt naar arbitraire configuraties.Het doel van deze thesis is om deze actie aan te wenden om te trachten het eerderaangehaalde resultaat uit [3] terug te vinden. Indien beide besproken methodes zoudenovereenstemmen in dit concrete geval, zou men de <strong>DBI</strong> actie ook kunnen gebruikenom te bestuderen wat er gebeurt in gebieden die de omgeving waarin de conformeveldentheoretische aanpak van toepassing is overschrijden. Dit doel werd niet bereikt,daar complicaties in de berekeningen, die in het toebedeelde tijdsbestek niet opgelostkonden worden, roet in het eten gooiden.2

Part IGeneral Introduction3

Chapter 1Introduction“Smokey, my friend, you’re entering a world <strong>of</strong> pain. Aworld <strong>of</strong> pain...”Coen BrothersThe Big Lebowski<strong>String</strong> theory is a field that is still in full blossoming, despite the fact that it has actuallybeen around for about four decades. It took its first steps rather slowly and faced a hardtime getting generally accepted for several reasons. In this introductory chapter, a bit<strong>of</strong> ink will be spilled in order to paint with very broad strokes the general idea <strong>of</strong> whatstring theory is all about, and in trying to retrace some <strong>of</strong> the most important steps itwent through from its inception until now.1.1 <strong>String</strong> theory in a nutshellUnlike the reference work <strong>of</strong> the same name [13], this nutshell will not comprise severalhundreds <strong>of</strong> pages and will, hopefully, be accessible to everyone.<strong>String</strong> theory sometimes seems like a giant box full <strong>of</strong> different types <strong>of</strong> buildingblocks. These building blocks allow one to construct entire worlds, and the name <strong>of</strong> thegame is <strong>of</strong> course to build a world that looks remarkably close to ours. However, due toan as <strong>of</strong> yet insufficient understanding <strong>of</strong> the fundamental behaviour <strong>of</strong> these buildingblocks, this task proves itself excruciatingly difficult.The Standard Model allowed for the unification <strong>of</strong> three <strong>of</strong> the four fundamentalforces at a quantum (field) level; electro-magnetism, and the weak and strong nuclearforces. Its success has been confirmed time and again by the tear inducingly beautifulagreement <strong>of</strong> its predicitions with experimental results. Which made its predictive powerall the more puzzling, given that the theory exhibited some undeniable shortcomings.For one, it relied on a great number <strong>of</strong> parameters (couplings, masses,...) which neededto be tuned by hand in order to agree with Nature. Furthermore, it did not incorporatethe fourth fundamental force: gravity. And last but certainly not least, the high energy4

CHAPTER 1. INTRODUCTION 5behaviour <strong>of</strong> the theory seemed to be far for from ideal, a regime at which gravity canno longer be ignored.All these problems suggested that a new, or at least improved, theory should exist,one that is more fundamental than the Standard Model. As far as the improvement<strong>of</strong> the Standard Model is concerned, two major developments occured. The first one isthe construction <strong>of</strong> the so-called Grand Unified Theories, which try to extend the gaugegroups <strong>of</strong> the Standard Model in order to further unify the theory. The other one issupersymmetry. But still, this proves to be insufficient.So rather than extending the SM, one could also envisage the second approach: tocreate a new fundamental theory.Figure 1.1: Taken from [23].<strong>String</strong> theory is a quantum theory that has the potential to incorporate all fourfundamental forces <strong>of</strong> Nature. Its novelty lies in the fact that it radically alters the concept<strong>of</strong> what an elementary particle is. All other theories except string theory considerelementary particles as being point particles; particles without any physical dimension.<strong>String</strong> theory raises the possibility that the fundamental constituents <strong>of</strong> matter mightwell have a physical dimension. It started out looking at fundamental constituentswhich extend in one spatial dimension closely resembling strings, explaining its name.The idea is that these strings can oscillate, and that different oscillations give rise todifferent particles. However, the original string soon became a “superstring,” and othermulti-dimensional objects also naturally appeared to inhabit the theory. Despite allthis, the name “string theory” remained; it simply gained weight.Complementing the possibility that it might be a quantum theory <strong>of</strong> gravity, stringtheory exhibits a number <strong>of</strong> other features that sets it apart from other theories. Forone, it contains only one dimensionfull parameter, namely the string length l s , the length<strong>of</strong> the fundamental constituent. All other “parameters” are dictated by the theory. Asa specific example, string theory demands the number <strong>of</strong> space-time dimensions to be10 (in superstring theory) in order to be consistent. It is the first theory to imposethe dimensionality <strong>of</strong> the space-time it lives in. Admittedly, 10 dimensions at first

CHAPTER 1. INTRODUCTION 6sight seems like six too many, which brings along the challenge <strong>of</strong> making those extradimensions “invisible.”Another compelling feature is that string theory requires supersymmetry. Also, yousimply can not ignore gravity; it is embedded in it from the very <strong>of</strong>fset.At one point, string theory appeared to be “diversified,” and loose its uniqueness.Several “flavours” <strong>of</strong> consistent superstring theories were known, and at first sight, theywere not related. However, it was found out afterwards that in fact all <strong>of</strong> these theorieswere related by an intricate play <strong>of</strong> dualities. As a consequence, string theory regainedits uniqueness.Of course, all is not peace and love in string land. The theory has its flaws. Forone, the exact description that gives rise to the physics as they are described by theStandard Model is as <strong>of</strong> yet still unknown. Also, giving the right mass to particlesis still a challenge. But most <strong>of</strong> all, string theory has failed so far to make even onesingle experimentally verifiable predicition. It has already made many predictions, butthey are simply vastly out <strong>of</strong> reach <strong>of</strong> modern day experiments. On the other handhowever, the development <strong>of</strong> string theory has given rise to many new insights into newmathematics and old physics alike. Moreover, the elegance and constrictiveness <strong>of</strong> itsformulation, as well as the many fundamental answers that seem to be lurking behindthe corner, easily justify its place at the top <strong>of</strong> modern theoretical physics.1.2 A brief history <strong>of</strong> string theoryThis section will summarily walk through some <strong>of</strong> the major steps in the history <strong>of</strong> stringtheory so far.1968: the seed from which string theory would eventually grow was sown in this year,during a time in which much study was devoted to the understanding <strong>of</strong> the stronginteractions, and understanding the hadronic spectrum and the many resonances fromstates with ever increasing spin that kept being discovered. Phenomenology showed thatmany <strong>of</strong> these resonances appeared to respect a linear behaviour between their mass andtheir spin,m 2 = J α ′ + α 0, (1.1)with m the mass, J the spin, α ′ the Regge-slope and α 0 the intercept. On the other hand,when considering four-particle scattering amplitudes (see Fig. 1.2) a duality betweenthe s- and t-channels appeared to exist, in that it could be shown (with the help <strong>of</strong>experimental data) that the amplitude for the s-channel and t-channel correspondedfor small enough values <strong>of</strong> s and t. This was called the “duality hypothesis,” and thissuggested that it should be possible to write down an amplitude A(s, t) that was invariantunder the exchange s ←→ t. Veneziano managed to write down such an amplitude bymaking use <strong>of</strong> Euler’s β-function,A(s, t) =Γ (−α(s)) Γ (−α(t)), (1.2)Γ(−α(s) − α(t))

Part IITheoretical Introduction11

Chapter 2Bosonic <strong>String</strong>s“If you’d like to know, I can tell you that in your universeyou move freely in three dimensions that you call space.You move in a straight line in a fourth, which you calltime, and stay rooted to one place in a fifth, which is thefirst fundamental <strong>of</strong> probability. After that it gets a bitcomplicated, and there’s all sorts <strong>of</strong> stuff going on in dimensionsthirteen to twenty-two that you really wouldn’twant to know about, even if you start from a position <strong>of</strong>thinking it’s pretty damn complicated in the first place.”Douglas AdamsHitchhiker’s Guide to the GalaxyWe will now turn to the basic element <strong>of</strong> string theory: the string itself. Just as everybodyelse, we will start at the beginning, meaning the bosonic string. Although thistheory is inadequate to describe our universe (at least without severe “modifications”), itis ideally suited for getting acquainted with a lot <strong>of</strong> concepts and terminology proper tostring theory, and many <strong>of</strong> the ideas present herein can be straightforwardly generalizedto the superstring case.Ultimately, our goal is to obtain the spectrum <strong>of</strong> a free bosonic string. To achievethis goal, we must first have an action. Once we have this action, we can compute theequations <strong>of</strong> motion (EOM), and we will see that different boundary counditions canbe imposed in order to solve these equations. Next we can quantize these equations.Expanding the quantized version <strong>of</strong> the EOM will deliver us what we are looking for.<strong>Analysis</strong> <strong>of</strong> this spectrum will tell us that we need a 26-dimensional space-time in orderfor this theory to possibly make any sense at all, but even then we will be left with twoserious issues, namely the presense <strong>of</strong> tachyons in the spectrum and the fact that thespectrum does not contain any fermions, which will only be solved in the next chapter.The idea behind string theory is that there exist a number <strong>of</strong> objects, most notablystrings and their generalised brothers, the p-branes, that live in an external,12

CHAPTER 2. BOSONIC STRINGS 13D-dimensional space-time (or background). p usually denotes the number <strong>of</strong> spatialdimensions only, hence, D > p. When these p-dimensional objects move in space-time,they describe a (p + 1)-dimensional world volume. On this world volume, one can, anddoes, define a set <strong>of</strong> coordinates, usually denoted by σ. Now, one can imagine standingsomewhere on this world volume, surrounded by the D-dimensional external world, andthrowing out anchors, one in each dimension <strong>of</strong> the external space-time. These anchorsdefine maps that translate the position on the world volume to a position in the externalspace-time, <strong>of</strong>ten referred to as the target space, and these are usually denoted by X.With both coordinate systems, that <strong>of</strong> the world volume and <strong>of</strong> space-time, are associatedmetrics. Also associated with a brane is a brane tension which gives a notion <strong>of</strong>how resistive this object is with respect to gravitation-like forces exerted upon it fromoutside. In a sense, this is nothing else but the mass per volume <strong>of</strong> a brane. In the case<strong>of</strong> the 0-brane, which is in fact a pointlike particle, the brane tension is the mass itselfsince the 0-brane has no physical dimension, and hence no volume. Starting from the1-brane, or string if you prefer, this tension thus becomes mass per length, mass persurface, etc. The metrics, maps and brane tension are all we need to construct an actionfor the brane. Since this chapter specializes to strings, in this case p = 1.2.1 ActionOur journey starts with the action <strong>of</strong> the bosonic string. Being one-dimensional, uponmoving around in space-time this object will describe a two-dimensional sheet, calledthe world sheet, analogous to the one-dimensional world line <strong>of</strong> a point particle. Weequip this world sheet with a coordinate system, the coordinates <strong>of</strong> which are usuallylabeled (σ 0 , σ 1 ) = (τ, σ), in which τ suggests that this coordinate can be tought <strong>of</strong> asthe equivalent <strong>of</strong> the space-time time t, and σ can be interpreted as a spatial dimension.The simplest action we can associate with this object is the following:∫ √S NG = −T dσdτ (ẊX′ ) 2 − Ẋ2 X ′2 . (2.1)In this equation, T represents the string tension, Ẋ = ∂ τ X and X ′ = ∂ σ X. The subscript“NG” stands for “Nambu-Goto,” the two people who first considered this action, makingit the Nambu-Goto action. The string tension T can be interpreted as the “mass perunit length” <strong>of</strong> the string, and hence it has mass dimension equal to two. Often thestring tension is written asT = 12πα ′, (2.2)where α ′ represents the Regge slope mentioned in the introductory chapter. One couldrewrite this action as∫S NG = −T√d 2 σ − det(η µν ∂ α X µ ∂ β X ν ), (2.3)with η µν representing the D-dimensional Minkowski metric, and the subscripts (α, β)refering to derivatives with respect to the world sheet coordinates. As such, for the

CHAPTER 2. BOSONIC STRINGS 14moment we consider space-time to be flat. Note that the minus sign preceding thedeterminant is only needed if a negative time-component is present, as is the case here,but if we were to e.g. consider a Euclidean space-time, we should disregard it. Furthernote that the expressionη αβ = η µν ∂ α X µ ∂ β X ν (2.4)defines a metric on the world sheet. This expression is <strong>of</strong>ten called the pullback η αβ <strong>of</strong>the tensor η µν , in this case the space-time metric, on the world sheet.Eq. 2.1 however is a classical action that does not lend itself very well to quantizationbecause <strong>of</strong> the presence <strong>of</strong> a square root. 1 Note that we have not yet made use <strong>of</strong> ametric explicitely defined on the world sheet (we only used the pullback from the externalspace-time). Doing so allows us to write a new action, called the Polyakov action orstring sigma model action. Writing this world sheet metric as h αβ , this action takes thefollowing form:S σ = − 1 2 T ∫d 2 σ √ −hh αβ η µν ∂ α X µ ∂ β X ν . (2.5)Here, h = det h αβ . The √ −h factor is actually needed to make the integral measure(d 2 σ √ −h) reparameterization invariant, a symmetry that will prove itself very usefullater on. The Nambu-Goto and Polyakov actions are equivalent at the classical level,which can be demonstrated by computing the equations <strong>of</strong> motion for h αβ , and usingthem to eliminate h αβ from the Polyakov action, which gives back the Nambu-Gotoaction, a point we will come back to later.Before going on, we will agree to the convention that indices (µ, ν, . . .) refer tospace-time, whilst indices (α, β, . . .) refer to the world sheet.2.2 SymmetriesNow that we have our action, we are interested in the symmetries <strong>of</strong> this action, especiallythe local ones, as we will need these to fix our metric (i.e. choose a gauge), a stepwe need to undertake if we wish to be able to quantize our theory. Keep in mind thatwe are still considering Minkowski space-time, which comes down to considering thefirst term in a perturbative expansion around a flat background. Going to the case <strong>of</strong>a curved background can be achieved by simply replacing η µν by g µν . 2 The symmetries<strong>of</strong> the Polyakov action are:Global symmetries• Poincaré transformations: the usual translations, rotations and boosts in spacetime,which are represented byδX µ = a µ νX ν + b µ ,δh αβ = 0,(2.6)1 In the case <strong>of</strong> a pointlike particle, it is also inadequate to describe massless particles, given thatT 0 = m.2 Note that this is true for the classical theory, but not for the quantum theory.

CHAPTER 2. BOSONIC STRINGS 15with a µ ν an infinitesimal Lorentz transformation parameter, and b µ an infinitesimaltranslation parameter, both constant.Local symmetries• Reparameterization invariance: this amounts to choosing different coordinateson the world sheet, and hence reparameterizing the maps X µ ; mathematicallyspeaking, this means applyingσ α −→ f α (σ, τ) = σ ′α ,h αβ (σ, τ) =∂f γ ∂f δ∂σ α ∂σ β h γδ(σ ′ , τ ′ ).(2.7)• Weyl transformations: in essence this means rescaling the metric with a coordinatedependent, and thus local, scaling factor,h αβ −→ e φ(σ,τ) h αβ ,δX µ = 0,(2.8)with φ(σ, τ) an arbitrary function <strong>of</strong> σ and τ. 3 These local symmetries allow us to gaugefix the world sheet metric so that the Polyakov action takes on a remarkably simple form,making the computation <strong>of</strong> the equations <strong>of</strong> motion particularly easy. First, we remarkthat a metric on a two-dimensional space has three independant components. As Eq.2.7 shows, reparameterizing two components requires two functions corresponding tothe new coordinates. Thus, we are free to choose these new coordinates such thath αβ = φ(σ, τ)η αβ , (2.9)where φ(σ, τ) represents a local conformal factor. But, Weyl invariance expressed inEq. 2.8 allows us to eliminate this local factor, leaving us with[ ] −1 0h αβ = η αβ = . (2.10)0 1This gauge choice is called the conformal gauge, because h αβ is trivially conformallyrelated to η αβ . The conformal gauge allows the Polyakov action (Eq. 2.5) to be rewrittenin the much simpler wayS = T ∫d 2 (Ẋ2 σ − X ′2) . (2.11)23 The exponential in Eq. 2.8 is not necessary, but makes it easiest to see that this is indeed a symmetry,as √ −hh αβ −→ e φ e −φ√ −hh αβ .

CHAPTER 2. BOSONIC STRINGS 162.3 Equations <strong>of</strong> motionFrom Eq. 2.11, we obtain the equations <strong>of</strong> motion for X µ .δS = T ∫ {}d 2 σ∂Ẋ22 ∂Ẋ (∂ τδX) − ∂X′2∂X ′ (∂ σ δX)= T ∫) ( )d 2 (σ{2∂ τ(ẊδX − 2 ∂ τ Ẋ δX − 2∂ σ X ′ δX ) + 2 ( ∂ σ X ′) }δX2∫= T d 2 σ ( ∂σX 2 − ∂τX 2 ) δX, (2.12)from which we conclude that∂ 2 X µ∂σ 2 − ∂2 X µ∂τ 2 = ∂ α ∂ α X µ = 0. (2.13)We thus obtain a two-dimensional wave equation for every map X µ .2.4 <strong>Bound</strong>ary conditionsLooking back at Eq. 2.12, we see that we have assumed, as is usual when varying anaction, that the total derivatives vanish. The total time (actually, τ) derivative is, asis customary, eliminated by choosing initial and final states for which the variations arezero, i.e.δX µ (σ, τ i ) = δX µ (σ, τ f ) = 0, (2.14)with τ i and τ f representing the initial and final time respectively. This leaves only theσ derivative, ∫− T dτ [ X µδX ′ µ | σ=π − X µδX ′ µ ]| σ=0 = 0, (2.15)in which we further assume that σ ∈ [0, π], for open and closed strings alike (a differencewe will quantify in a second). Now, there are several ways to make this condition fit,and these different possibilities mathematically define different types <strong>of</strong> strings.2.4.1 Closed stringsA first possibility is to assume thatX µ (σ, τ) = X µ (σ + π, τ) . (2.16)This would <strong>of</strong> course imply that the string is periodic, or closed, and as a consequence,since we have just identified points whose σ values differ by an integer multiple <strong>of</strong> π,that alsoδX µ (σ, τ) = δX µ (σ + π, τ). (2.17)Hence, this condition defines a closed string.

CHAPTER 2. BOSONIC STRINGS 172.4.2 <strong>Open</strong> stringsDirichlet boundary conditionsAnother possibility is to assume that δX µ = 0, which would imply that we are dealingwith an open string whose endpoints are fixed:X µ | σ=0 = X µ 0 ; X µ | σ=π = X µ π. (2.18)This defines an open string with Dirichlet boundary conditions. Since we are now demandingthat the endpoints <strong>of</strong> the string correspond to some specific points in space,we can no longer hope that this string satisfies Poincaré invariance. For this reason,Dirichlet boundary conditions are not even mentioned in older references like e.g. [9]and [14], until they eventually turned out to be <strong>of</strong> great interest, as we will found outourselves in Chapter 5.Note that for this condition µ ≠ 0, since τ ∼ = t and Eq. 2.18 can be restated asNeumann boundary conditions∂X µ∂τ∣ = 0. (2.19)σ=0,πFinally, we could impose that the σ derivatives vanish in the endpoints <strong>of</strong> an open string,∂X µ∂σ (σ = 0, τ) = ∂X µ(σ = π, τ) = 0. (2.20)∂σThis defines what is called an open string with Neumann boundary conditions. It is<strong>of</strong>ten stated that “no momentum flows <strong>of</strong> the end <strong>of</strong> an open string with Neumannboundary conditions,” which is indeed what Eq. 2.20 expresses. This might makeyou wonder where the momentum that can flow <strong>of</strong>f the ends <strong>of</strong> a Dirichlet string goesto. Anticipating what will come, the endpoints <strong>of</strong> a Dirichlet string are attached to aphysical object called a D-brane, and hence momentum gets transferred to this objectand vice-versa.Note that in the case <strong>of</strong> an open string, one could impose Neumann boundary conditionsin some directions, and Dirichlet boundary conditions in the others.2.5 Solutions and expansionsNow that we have our equations <strong>of</strong> motion for the bosonic string, it is time to lookfor solutions to it. To this end, it will prove to be useful to switch to a new set <strong>of</strong>coordinates, called light-cone coordinates or light-cone gauge. These are defined byeasily explaining their name. As a consequence, usingσ ± = τ ± σ. (2.21)∂ ± =∂∂σ ±,

CHAPTER 2. BOSONIC STRINGS 18we see that∂ + = 1 2( ∂τ ∂∂σ + ∂τ + ∂σ )∂∂σ + ,∂σ= 1 2 (∂ τ + ∂ σ ) . (2.22)Similarly,∂ − = 1 2 (∂ τ − ∂ σ ) . (2.23)For the metric, we compute thatand analogous for η −− , andη ++ = ∂σµ∂σ + ∂σ ν∂σ +η µν,= (∂ + σ) 2 − (∂ + τ) 2 ,= 1 4 − 1 4 ,= 0, (2.24)η ±∓ = (∂ + σ)(∂ − σ) − (∂ + τ) (∂ − τ),= − 1 4 − 1 4 ,= − 1 2 , (2.25)leaving us with [ ]η++ η +−= − 1 [ 0 1η −+ η −− 2 1 0]. (2.26)The nice thing about these coordinates is that they allow us to write the equations <strong>of</strong>motion expressed in Eq. 2.12 as follows:∂ + ∂ − X µ = 0, (2.27)which immediatly suggests the general solutionX µ (σ, τ) = X µ (R σ− ) + X µ (L σ+ ) . (2.28)X µ R,Lare called the right- and left-moving modes <strong>of</strong> the string respectively.Things get a bit messy when we want to expand these modes, because we have tolook at it on a case per case basis, depending on what boundary conditions are appliedto the string under consideration.

CHAPTER 2. BOSONIC STRINGS 192.5.1 Closed string expansionIf we assume closed string boundary conditions, these modes can be expanded as follows:X µ R = 1 2 xµ + 1 2 l2 sp µ (τ − σ) + i 2 l ∑ 1sn αµ ne −2in(τ−σ) , (2.29)n≠0X µ L = 1 2 xµ + 1 2 l2 sp µ (τ + σ) + i 2 l ∑ 1sn ˜αµ ne −2in(τ+σ) . (2.30)Here, we see the introduction <strong>of</strong> the string length scale l s , related to the Regge slope α ′and string tension T by the relations 4n≠012 l2 s = α ′ ; T = 1πls2 , (2.31)the center <strong>of</strong> mass momentum p µ , and the center <strong>of</strong> mass position x µ . More importantlythough, we also see the introduction <strong>of</strong> positive and negative modes, which in thequantum version <strong>of</strong> the story will become raising and lowering operators. Note that X µshould be real, 5 and that as a consequenceα µ −n = (αµ n) ∗ ; ˜α µ −n = (˜αµ n) ∗ . (2.32)The fact that x µ and p µ do indeed represent the center <strong>of</strong> mass position and momentumcan be obtained as follows. First note that∂X µ R (σ− )∂τ= ∂Xµ R (σ− ) ∂σ −∂σ − ∂τ = ∂ σ −Xµ R , (2.33)and analogous for X µ L . So defining α µ 0 = ˜αµ 0 = (l s/2)p µ (2.34)so as to allow us to “complete the sum” upon differentiation, we find thatẊ µ R = l sẊ µ L = l s+∞∑n=−∞+∞∑n=−∞α µ ne −2in(τ−σ) , (2.35)˜α µ ne −2in(τ+σ) . (2.36)From this, and using the usual definition <strong>of</strong> the conjugate momentum, we see thatP µ = ∂L∂Ẋµ= T∫ π0dσẊµ = T∫ π0)dσ(ẊµL + Ẋµ R= p µ , (2.37)4 Some references like e.g. [13] define ls 2 = α ′ , inducing subsequent changes in several formulae withfactors <strong>of</strong> √ 2, 2 and/or inverses there<strong>of</strong>.5 We are, after all, mapping from and to real spaces.

CHAPTER 2. BOSONIC STRINGS 20which confirms our claim for p µ . Furthermore, simply applying the definition <strong>of</strong> anaverage to X µ (σ), we see that1π∫ π0dσX µ (σ, τ) = x µ + l 2 sp µ τ, (2.38)indicating that x µ is the center <strong>of</strong> mass position at time τ = 0. Note that x µ and p µare arbitrary, i.e. if we wanted to we could explicitely fill them in, thereby specifying aninitial position and momentum for our string.2.5.2 <strong>Open</strong> string expansionA feature present in all open string expansions is that only one set <strong>of</strong> expansion coefficientsα is present. Before giving the expansions, this can already be understoodby realizing that whilst the closed strings allow oscillations in two directions, in thecase <strong>of</strong> open strings left- and right-movers need to combine into standing waves due toreflections at their endpoints, hence leaving only one set <strong>of</strong> coefficients.Dirichlet boundary conditionsThe wave equations for an open string satisfying Dirichlet boundary conditions can beexpanded as∑X µ (σ, τ) = x µ + w µ 1σ − l sn αµ ne −inτ sin(nσ), (2.39)with again µ ≠ 0. We need sin(nσ), and not cos (nσ), in order to accomodate thevanishing <strong>of</strong> the oscillators in the endpoints. The constant w µ is called the windingnumber <strong>of</strong> the string. Note thatn≠0X µ (σ = π, τ) − X µ (σ = 0, τ) = w µ π, (2.40)and hence w µ encodes the distance between both endpoints. We will come back to thiswhen we will consider T-duality and D-branes.Neumann boundary conditionsFor an open string satisfying Neumann boundary conditions, the expansion readsX µ (σ, τ) = x µ + l 2 sp µ τ + il s∑n≠01n αµ ne −inτ cos (nσ). (2.41)Note that this time we need cos (nσ), so that the ∂ σ derivatives will vanish at theendpoints. Also note that since no momentum can flow <strong>of</strong> the ends <strong>of</strong> the string, we canagain talk about a center <strong>of</strong> mass momentum p µ .This time, we can define the zero-mode asα µ 0 = l sp µ . (2.42)

CHAPTER 2. BOSONIC STRINGS 21We will not consider the more exotic case <strong>of</strong> a string which has a Dirichlet boundarycondition at one end, and Neumann boundary condition on the other end, which istreated in e.g. §2.3.2 in [13].2.6 QuantizationSo far, everything we have done was purely classical. In order to quantize the theory,three basic approaches exist, <strong>of</strong> which we only consider the following two:• covariant approach: the usual QFT quantization, in which the expansion coefficients<strong>of</strong> the expanded fields get promoted to quantum operators satisfying commutatorrelations that are generalized from the Poisson brackets satisfied by theclassical fields; its advantage is that it manifestly preserves Lorentz invariance,• light-cone quantization: a specific change <strong>of</strong> variables is made that allows oneto solve the constraint equations at a classical level instead <strong>of</strong> having to imposethem, and quantize the remaining theory; the drawback is that manifest Lorentzinvariance is lost, the advantage is the inherent removal <strong>of</strong> negative-norm states.The third method is path integral quantization, and is technically vastly more complicated.Interested parties are referred to e.g. [13] §3.5, [14] §3.4 and [16] §2.5. It isworth mentioning that whenever comparison between these methods was possible, theyall agreed.2.6.1 Covariant approachThe idea is pretty simple: compute the classical Poisson brackets (P.B.’s) <strong>of</strong> the expandedX µ ’s, translate those to the P.B.’s <strong>of</strong> the expansion coefficients, replace theseP.B.’s by quantum commutators, throw in a factor <strong>of</strong> i, et voilà, you are done. Let usgo through the motions one by one.Again defining the momentum conjugate to X µ asP µ (σ, τ) = ∂L∂Ẋµ= TẊµ , (2.43)and recalling that the Poisson brackets <strong>of</strong> two arbitrary functions x(q, p),y (q, p) aredefined by[ ∂x ∂y{x, y} P.B. =∂q ∂p − ∂x ]∂y, (2.44)∂p ∂qwith q representing coordinates and p representing momenta, we find that{P µ (σ, τ),P ν ( σ ′ , τ )} P.B. = 0, (2.45){X µ (σ, τ),X ν ( σ ′ , τ )} P.B. = 0, (2.46){P µ (σ, τ),X ν ( σ ′ , τ )} P.B. = ηµν δ ( σ − σ ′) , (2.47){Ẋ (σ, τ),Xν ( σ ′ , τ )} P.B. = T −1 η µν δ ( σ − σ ′) , (2.48)

CHAPTER 2. BOSONIC STRINGS 22which are the usual equal time classical Poisson brackets. This can be translated to theP.B.’s <strong>of</strong> the modes by inserting the expansions found earlier (Eqs. 2.29, 2.30, 2.39 and2.41), and results in{α µ m, α ν n} P.B. = imη µν δ m+n,0 , (2.49){˜α µ m, ˜α ν n} P.B. = imη µν δ m+n,0 , (2.50){α µ n, ˜α ν n} P.B. = 0. (2.51)Of course, the two last equations are only applicable to the case <strong>of</strong> the closed string.The next step would be to replace these classical P.B.’s with quantum mechanicalcommutation relations usig the prescriptionThus we obtain{. . .,...} P.B. −→ i [. . . , . . .] . (2.52)[α µ m, α ν n] = [˜α µ n, ˜α ν n] = mη µν δ m+n,0 ; [α µ m, ˜α ν n] = 0. (2.53)Just as in quantum mechanics, these operators are normalized following the definitiona µ m = 1 √ mα µ m ; a µ†m = 1 √ mα µ −m with m > 0. (2.54)Essentialy, this leaves us with the algebra <strong>of</strong> the harmonic oscillator operators,[ ] [ ]a µ m, a ν†n = ã µ m, ã ν†n = η µν δ m,n with m, n > 0. (2.55)This immediatly alerts us to a serious problem, namely[ ]a 0 m, a 0†m = −1, (2.56)which demonstrates that the timelike mode (µ = 0) is a very naughty boy because itgives rise to negatively normed states, e.g.〈0|a 0 ma 0†m|0〉 = −1,in which we assumed that 〈0|0〉 = 1, and in which |0〉 represents the ground state, i.e.the state that is annihilated by all lowering operators,α µ m|0〉 = 0 ∀ m > 0. (2.57)This, and other negative norm states, are called ghosts, and we will need to find a wayto get rid <strong>of</strong> them if we want our theory to make any sense.

CHAPTER 2. BOSONIC STRINGS 232.6.2 Energy-momentum tensor and HamiltonianWe have come to a point where we can no longer ignore the energy-momentum tensorT αβ , defined asT αβ = − 2 1 δL√ σ, (2.58)T −h δhαβ or if you prefer, as the variation <strong>of</strong> the Lagrangian (density) with respect to the (worldsheet) metric. In this equation, L σ represents the Lagrangian density as it appears inthe Polyakov action defined in Eq. 2.5. Using the Polyakov action andone finds that∫δSδh αβ = −1 2 T= − 1 2 T √ ∫−h{( √ ) ∂ −hd 2 σ∂h αβd 2 σδ √ −h = − 1 2√−hhαβ δh αβ ,h γδ η µν ∂ γ X µ ∂ δ X ν + √ }−hη µν ∂ α X µ ∂ β X ν δh αβ{− 1 }2 h αβh γδ ∂ γ X µ ∂ δ X µ + ∂ α X µ ∂ β X µ δh αβ , (2.59)implying thatT αβ = ∂ α X · ∂ β X − 1 2 h αβh γδ ∂ γ X · ∂ δ X. (2.60)Demanding that the field equations for the world sheet metric,δLδh αβ= 0, are satisfied,we need to impose thatT αβ = 0. (2.61)These equations are called the constraint equations, as they lay extra constraints on thesolutios <strong>of</strong> the EOM for X µ . In the particular case <strong>of</strong> bosonic string theory, they also gounder the name <strong>of</strong> Virasoro constraints, for reasons that will become clear in a second.We have seen that one <strong>of</strong> the local symmetries <strong>of</strong> the world sheet is Weyl invariance. Thissymmetry has a side effect, namely it induces the tracelessness <strong>of</strong> the energy-momentumtensor, i.e. h αβ T αβ = 0. To show this, consider a general action S (h αβ , φ i ), where φ irepresents some fields satisfying the field equations δL = 0. Supposing that under anδφ iinfinitesimal Weyl rescaling with parameter Λ these transform ash αβ −→ h αβ + 2Λh αβ ; φ i −→ φ i + d i Λφ i , (2.62)where d i represents the conformal weight (see Chapter 4) <strong>of</strong> φ i , and assuming the actionto be scale invariant, we find that∫ {0 = δS = d 2 σ 2 δL h αβ + ∑ }δLd i φ i Λ. (2.63)δh αβ δφi iDue to the field equations <strong>of</strong> φ i , only the first term is non-trivial. Using Eq. 2.58, wefind that∫0 = −T d 2 σ √ )−h(T αβ h αβ , (2.64)

CHAPTER 2. BOSONIC STRINGS 24from which the claimed result follows. Note that Eq. 2.8 shows that the conformalweight <strong>of</strong> the fields X µ is zero, and hence in this case the result follows even withoutthe use <strong>of</strong> the equations <strong>of</strong> motion.As was mentioned earlier, the equations <strong>of</strong> motion for h αβ , resulting in T αβ , can beused to show the classical equivalence between the NG and Polyakov actions. First,using Eqs. 2.60 and 2.61, remark that− det(∂ α X · ∂ β X) = − 1 ) 2(h4 h γδ ∂ γ X · ∂ δ X⇓√− det(g µν (X) ∂ α X µ ∂ β X ν ) = 1 √ ( )−h h γδ ∂ γ X · ∂ δ X . (2.65)2Plugging this result right into Eq. 2.5 immediatly gives back Eq. 2.1.In conformal gauge, the constraint equations readand in light cone coordinates they translate toT 01 = T 10 = Ẋ · X′ = 0, (2.66)T 00 = T 11 = 1 2 (Ẋ2 + X ′2) = 0, (2.67)T ++ = ∂ + X µ ∂ + X µ = 0, (2.68)T −− = ∂ − X µ ∂ − X µ = 0, (2.69)T +− = T −+ = 0. (2.70)This last condition expresses the vanishing <strong>of</strong> the trace in light-cone gauge, η αβ T αβ =0 = T +− + T −+ .Virasoro generatorsInserting our mode expansions for X µ for the closed string into the above expresion forT ++ , we find thatT ++ = ∂ + X L · ∂ + X L( +∞) (∑∑ +∞= ls2 ˜α me µ −2im(σ+ ) ·= l 2 s= l 2 sm=−∞( +∞∑+∞∑m=−∞ n=−∞+∞∑( +∞∑m=−∞ n=−∞n=−∞)˜α m · ˜α n e −2i(n+m)(σ+ ))˜α m−n · ˜α n e −2i(m)(σ+ ))˜α ne µ −2in(σ+ )where in the last step, we made use <strong>of</strong> the fact that the summing index runs over aninfinite range <strong>of</strong> values. The computation for T −− is strictly analogous, and thus we see

CHAPTER 2. BOSONIC STRINGS 25thatin which we usedT −− = 2l 2 sT ++ = 2l 2 sL m = 1 2˜L m = 1 2+∞∑m=−∞+∞∑m=−∞+∞∑n=−∞+∞∑n=−∞L m e −2im(τ−σ) , (2.71)˜L m e −2im(τ+σ) , (2.72)α m−n · α n , (2.73)˜α m−n · ˜α n . (2.74)These entities are called the Virasoro generators, and they obey the so-called Virasoroalegbra, to which we will come back in a second.For open strings, the Virasoro constraints look a little different, but they still allowfor the same combination <strong>of</strong> modes to be separated, and hence, the Virasoro generatorsremain the same. If this seems a bit awkward at first, recall that for open strings leftandright-movers need to combine into standing waves. This does however still allowthe expansion to be separated into left- and right-movers, and hence the derivation <strong>of</strong>the Virasoro constraints remains strikingly resemblant to that <strong>of</strong> the closed string case.Anyhow, we are mostly interested in the Virasoro generators, not the constraints.HamiltonianUsing the Virasoro generators, one can easily write the Hamiltonian asH = 2(L 0 + ˜L)0 =in the case <strong>of</strong> the closed string, andH = L 0 = 1 2+∞∑n=−∞+∞∑n=−∞(α −n · α n + ˜α −n · ˜α n ) (2.75)α −n · α n (2.76)for an open string. One obtains this result by first considering the definition <strong>of</strong> theHamiltonian, ∫ πH =(Ẋµ P µ − L)dσ = T ∫ π (Ẋ2 + X ′2) dσ. (2.77)02 0The next step is <strong>of</strong> course to compute the derivatives <strong>of</strong> the X µ ’s. Considering the case<strong>of</strong> the closed string, using the expansions shown in Eqs. 2.29 and 2.30 and not forgetting

CHAPTER 2. BOSONIC STRINGS 26to use Eq. 2.34, this readsẊ µ R = l s∑α µ ne −2in(τ−σ) , (2.78)Ẋ µ L = l s∑˜α µ ne −2in(τ+σ) , (2.79)X µ′R = −l s∑α µ ne −2in(τ−σ) , (2.80)X µ′L = l s∑˜α µ ne −2in(τ+σ) . (2.81)Next, we compute the square <strong>of</strong> these derivatives.)Ẋ 2 = Ẋ2 R + Ẋ2 L + 2(ẊR · Ẋ L⎧( )⎨2 ( ) ⎫2⎬ = ls∑α 2 µ ⎩ne −2in(τ−σ) +∑˜α ne µ −2in(τ+σ) ⎭{( ) ( )}+ 2ls∑α 2 n e −2in(τ−σ) ·∑˜α n e −2in(τ+σ)(2.82)(X′ ) 2 ( )= X′ 2 ( )R + X′ 2 (L + 2 X′R · X L)′⎧( )⎨2 ( ) ⎫2⎬ = ls∑α 2 µ ⎩ne −2in(τ−σ) +∑˜α ne µ −2in(τ+σ) ⎭{( ) ( )}− 2ls∑α 2 n e −2in(τ−σ) ·∑˜α n e −2in(τ+σ)(2.83)When summing Eqs. 2.82 and 2.83, we see that the crossterms will cancel each other,and that the squared terms will just gain a coefficient <strong>of</strong> two:⎧( )⎨2 ( ) ⎫2⎬ Ẋ 2 + X ′2 = 2ls∑α 2 µ ⎩ne −2in(τ−σ) +∑˜α ne µ −2in(τ+σ) ⎭ . (2.84)A general term in these squared sums can be written as(α m · α n )e −2i(τ−σ)(m+n) , (2.85)and this expression is integrated in Eq. 2.77. Recalling that∫ π0dσ e 2iσ(m+n) = πδ m+n , (2.86)

CHAPTER 2. BOSONIC STRINGS 27we are left withH = Tπl 2 s∑(α −n · α n + ˜α −n · ˜α n ). (2.87)Further using Eq. 2.31, this indeed resolves to Eq. 2.75, as we wanted to assure ourselves<strong>of</strong>. The derivation <strong>of</strong> the open string Hamiltonian goes by similarly. This, however,is a valid classical result, but as always, quantum mechanically we should deal withan ordering ambuigity. However at the moment we can ignore this complication, andremaining in the classical realm we can define an expression for the mass <strong>of</strong> a string as afunction <strong>of</strong> its oscillations. To this end, recall the light-cone gauge constraint equations,expressed in Eqs. 2.68 - 2.70, and remark that this implies that L m = ˜L m = 0 for allm. In other words, this should also hold for the zero-mode, and thus we find thatL 0 = 1 2∑α −n · α n = 1 2 α2 0 + ∑ n>0α −n · α n = 0. (2.88)and analogous for ˜L m (remember that α 0 = ˜α 0 ). This bears a nice surprise, because alook at Eqs. 2.31 and 2.34 reveals that for the closed string case12 α2 0 = 1 4 α′ p µ p µ . (2.89)For the open (Neumann) string, we have to use Eq. 2.42 revealing that12 α2 0 = α ′ p µ p µ . (2.90)Using the relativistic mass-shell condition M 2 = −p 2 , we find that for an open stringα ′ M 2 = ∑ n>0α −n · α n . (2.91)For the closed string, one should take into account contributions from both left- andright-movers, so thatα ′ M 2 = 2 ∑ n>0(α −n · α n + ˜α −n · ˜α n ) . (2.92)2.6.3 Virasoro algebra and physical statesClassically, the Virasoro generators satisfy the Virasoro algebra,{L m , L n P.B. = i (m − n)L m+n , (2.93){˜Lm , ˜L}n = i (m − n) ˜L m+n , (2.94)P.B.{L m , ˜L}n = 0. (2.95)P.B.

CHAPTER 2. BOSONIC STRINGS 28This algebra can be derived by using the commutation relations for the expansion modesexpressed in Eqs. 2.49 - 2.51. Quantum mechanically, things get a bit messier because<strong>of</strong> the earlier noted ordering ambiguity, and the algebra becomes[L m , L n ] = (m − n)L m+n + c12 m( m 2 − 1 ) δ m+n . (2.96)The derivation <strong>of</strong> this result is somewhat tricky, and can be found in full glory in e.g.the Appendix to Chapter 3, p. 54-55 in [14], or via an alternative approach in [9], p.80-81.As is usual in quantum theories, we will order our operators using the normalorderingconvention <strong>of</strong> “lowering operators to the right and raising operators to theleft,” resulting inL m = 1 2+∞∑n=−∞: α m−n · α n : . (2.97)We should be careful though. Since we find ourselves in the beloved and charmingpresence <strong>of</strong> gravity (on the world sheet), we can no longer ignore the extra constant thatresults from the normal ordering prescription as we did in the case <strong>of</strong> QFT. However,since all operators commute except those constituting L 0 , this ordering ambiguity isonly relevant for L 0 , and we take this into account by replacing L 0 by (L 0 − a) wheneverneeded, where a represents the as-<strong>of</strong>-yet undetermined ordering constant. 6This constant also has repercussions for the mass <strong>of</strong> states. We demand that for aphysical state |φ〉L m |φ〉 = ˜L m |φ〉 = 0 m > 0, (2.98))(L 0 − a) |φ〉 =(˜L0 − a |φ〉 = 0. (2.99)The second condition is simply the mass-shell condition. The first condition can be understoodas the quantum translation <strong>of</strong> the classical vanishing <strong>of</strong> the energy-momentumtensor T αβ . Quantum mechanically, our first guess would be to impose thatT αβ |φ〉 = 0, (2.100)however this would imply that all operators annihilate physical states. This would leaveus a little embarassed, as little physics would survive. The next best thing we can do isdemand that〈φ|T αβ |φ〉 = 0. (2.101)Remembering that in the mean time we normal-ordered our Virasoro generators, we seethat this time around the lowering operators can destroy |φ〉, while the raising operatorscan destroy 〈φ|, which leaves a fully consistent spectrum <strong>of</strong> physical states. This isanalogous to the statement expressed in Eq. 2.98.6 Basically, we are simply rewriting L 0 = `12P: α −n · α n : +a´ as L 0 − a = 1 2P: α −n · α n :.

CHAPTER 2. BOSONIC STRINGS 29Moving on, we see that for the closed string, Eq. 2.99 can be combined into(L 0 − ˜L 0)|φ〉 = 0, (2.102)a condition which bears the name <strong>of</strong> level-matching condition, since it implies that thenumber <strong>of</strong> left- and right-movers should be equal. To see this, we identifyN = ∑ α −n · α n ; Ñ = ∑n>0n>0˜α −n · ˜α n , (2.103)and recognize N and Ñ as variations <strong>of</strong> the well known number operator from QFT. Wealso see that Eq. 2.99 tells us that for a physical open string state( )∑α ′ M 2 |φ〉 = α −n · α n − a |φ〉. (2.104)n>0For the open string vacuum, this relation becomesα ′ M 2 |0〉 = −a|0〉,or in other words, the mass squared <strong>of</strong> the open string vacuum is proportional to −a.For the closed string vacuum, this becomes)α ′ M 2 |0〉 = 2(N + Ñ − 2a = −4a|0〉.This raises our curiosity as to what the value <strong>of</strong> a is exactly, because if it is strictlypositive, we have a serious problem. In order to find out what a is equal to, it ishowever much more convenient to switch to a different gauge.2.6.4 Light-cone gauge quantization and spectrumIn order to get started, we introduce light-cone coordinates in space-time. To this end,we defineX ± = √ 1 (X 0 ± X D−1) . (2.105)2The other D − 2 coordinates remain unchanged, and are denoted by X i with i ∈{1, . . .,D − 2}. The inner product now takes on the formX · Y = X µ Y µ = −X + Y − − X − Y + + ∑ iX i Y i . (2.106)Next, we notice that by gauge fixing h αβ = η αβ , we actually did not use all thefreedom we had at our disposal. In fact, after performing this gauge fix, so-calledresidual gauge freedom lurks around the corner, which is a consequence <strong>of</strong> the fact thattransformations <strong>of</strong> the form∂ α ξ β + ∂ β ξ α = Λη αβ , (2.107)

CHAPTER 2. BOSONIC STRINGS 30World SheettheoryConformal GaugeQuantizeLigth-ConeGaugeVirasoroConstraintsQuantize<strong>String</strong>sFigure 2.1: Conformal vs. light-cone quantization.with ξ α an infinitesimal reparameterization parameter and Λ and infinitesimal Weylrescaling parameter, leave our physics, and in particular our previous gauge choice,unchanged. To see this, first note that we could rewrite the local world sheet symmetries(Eqs. 2.7 and 2.8) in infinitesimal form asδh αβ = ξ γ ∂ γ h αβ − ∂ γ ξ α h γβ − ∂ γ ξ β h αγ , (2.108)δh αβ = Λh αβ , (2.109)where the first line represents an infinitesimal reparameterization, and the second linean infinitesimal Weyl rescaling. Replacing h αβ with η αβ , we see that we could chooseparameters such that Eq. 2.107 is indeed satisfied, i.e. we could perform a reparameterizationthat in fact has the same effect as a Weyl rescaling, and then cancel thisrescaling by applying the “inverse” Weyl rescaling.Defining ξ ± = ξ 0 ± ξ 1 , and using our old friends σ ± , Eq. 2.107 tells us thatwhich can be solved by stating that∂ + ξ + = ∂ − ξ + = 0,∂ − ξ − = ∂ + ξ − = 0,ξ + = ξ + ( σ +) ; ξ − = ξ − ( σ −) . (2.110)This implies that this residual gauge freedom allows us to apply reparameterizations <strong>of</strong>the formσ ± −→ ˜σ ± ( σ ±) , (2.111)and in particular τ = 1 2 (σ+ + σ − ) and σ = 1 2 (σ+ − σ − ) get transformed according to

CHAPTER 2. BOSONIC STRINGS 31the general form˜τ = 1 +[˜σ ( σ +) + ˜σ − ( σ −)] ,2(2.112)˜σ = 1 +[˜σ ( σ +) − ˜σ − ( σ −)] .2(2.113)You might be wondering why this is any reason to get excited about, but. . . Lookingback at Eq. 2.28, remembering that we got there from Eq. 2.27, and realizing that weare dealing with the “inverse” situation here, 7 we conclude that ˜τ may be any arbitrarysolution to the wave equation(∂2σ − ∂τ2 )˜τ = 0. (2.114)This is great news, because we know from the conformal gauge Virasoro constraintexpressed in Eq. 2.67 that the X µ ’s satisfy this very equation, and hence we can choose˜τ to be proportional to any <strong>of</strong> the X µ ’s. In particular, we can chooseor if you prefer,˜τ = 1l 2 sp + (X + (˜σ, ˜τ) − x +) , (2.115)X + (˜σ, ˜τ) = x + + l 2 sp +˜τ,which corresponds to setting the α + n (and ˜α + n ) to 0 for n ≠ 0 in the mode expansion forX + . Since the oscillators for X + have been put to 0 this way, this also means we canno longer excite our string in that direction! But that is not all. Since all components<strong>of</strong> T αβ are zero anyway, we can combine them and rewrite the Virasoro constraints inEqs. 2.66 and 2.67 as0 = T 00 + T 11 + T 01 + T 10 = Ẋ2 + 2Ẋ · X′ + X ′2 =(Ẋ + X′ ) 2, (2.116)0 = T 00 + T 11 − T 01 − T 10 = Ẋ2 − 2Ẋ · X′ + X ′2 =(Ẋ − X′ ) 2. (2.117)In light-cone gauge this becomes(Ẋ ± X′ ) 2= Ẋ 2 + X ′2 ± 2Ẋ · X′= −2Ẋ+ Ẋ − + Ẋi Ẋ i − 2X +′ X −′ + X i′ X ′ i∓ 2Ẋ+ X −′ ∓ 2X +′ Ẋ − ± 2Ẋi X ′ i0 = −2l 2 sp + Ẋ − +(Ẋi ) 2+(Xi′ ) 2∓ 2l2s p + X −′ ± 2Ẋi X ′ i⇓[Ẋ− ± X −′] = 12l 2 sp + (Ẋi ± X i′) 2. (2.118)7 More specifically, we have an equation <strong>of</strong> the same form as Eq. 2.27, and thus the solutions to thisequation will be <strong>of</strong> the same form as Eq. 2.28.

CHAPTER 2. BOSONIC STRINGS 32This gives us two equations relating the mode expansions <strong>of</strong> X − to those <strong>of</strong> the X i ’s, arelation that can be solved explicity, and for the case <strong>of</strong> the open string the result is(αn − = 1 1l s p + 2D−2∑∞∑i=1 m=−∞: α i n−mα i m : −aδ n,0). (2.119)For the closed string, an analogous relation for ˜α n − exists. Notice the ordering constanta again. The important point to note, is that now there are no independent oscillatorsleft for X − , so here again, we can no longer excite our string in the X − direction! Thatis very good news indeed, because looking back at Eq. 2.56, and keeping in mind thatdue to our freshly introduced light-cone coordinates our metric has somehow changed abit, 8 we now see that we are no longer able to create ghost states, even if we wanted todo so. In other words, the light-cone gauge is manifestly free <strong>of</strong> ghost states.Let us now turn to the mass operator. The generic expression M 2 = −p µ p µ nowbecomesD−2∑M 2 = −p µ p µ = 2p + p − − p 2 i . (2.120)In order to work this out some more, let us focus on the case <strong>of</strong> open (Neumann) strings,and recall that in that caseα µ 0 = l sp µ .Hence, using p − = (l s ) −1 α0 − and Eq. 2.119, we compute that{ [(D−2∑D−2)2p + p − − p 2 i = 2p+ 1 ∑ ∑l s l s p + α−nα i ni + 1 2i=1i=1where we redefinedD−2∑− 1 ls2 i=1[(= 2 D−2 ∑ ∞∑ls2 i=1 n=1(αi0) 2,α i −nα i nn>0)− a],i=1D−2∑+∞∑i=1 n=−∞(αi0) 2]M 2 = 2 ls2 (N − a), (2.121)N =D−2∑i=1 n=1∞∑α−nα i n.iWe see that the mass-shell condition in light-cone gauge is identical to the one in conformalgauge (Eq. 2.104), except for the fact that we now simply have two directions lessin which we can excite our string. Hence, our groundstate still has α ′ M 2 |0〉 = −a|0〉.8 More specifically, we get g +− = g −+ = −1, and g ij = δ ij for i, j ∈ {1, . . . , D − 2}. But the twonegatively valued entries correspond to directions in which we have no oscillators, and hence we can nolonger create ghosts.− a}

CHAPTER 2. BOSONIC STRINGS 33If we now take a look at the first excited state <strong>of</strong> our string, noting again that in thelight-cone gauge we can only excite transversely, we notice that this state,α i −1|0〉, (2.122)should belong to a (D − 2)-component vector representation <strong>of</strong> the rotation groupSO(D − 2). The mass <strong>of</strong> this state equalsα ′ M 2 ( α i −1|0〉 ) = (1 − a) ( α i −1|0〉 ) . (2.123)But, a state belonging to a vector representation <strong>of</strong> SO(D −2) should, if Lorentz invarianceis conserved, be massless.Since we want Lorentz invariance to be conserved, we areforced to “choose” a = 1.As a bonus, now that we know the value <strong>of</strong> a, we can determine, be it in a nonrigourousway, the dimension <strong>of</strong> space-time. Consider the non-normal-ordered operatorL 0 = 1 ∑ D−2 ∑ +∞2 i=1 −∞ αi −nαn. i Upon applying the normal-ordering prescription, we pickup a normal-ordering constant because <strong>of</strong> the commutators <strong>of</strong> the modes,12D−2∑+∞∑i=1 −∞: α i −n · α i n :=D−2∑∑α−n i · αn i + 1 2i=1 n>0D−2∑i=1( )αi 2 1∞0 −2 (D − 2) ∑n, (2.124)and as we have just found out, this constant should equal −1 (remember, we wrote“L 0 − a”). The trick to apply in order to obtain this result, is to use the Riemann zetafunction,∞∑ζ (s) = n −s , (2.125)n=1defined for any s ∈. The nice thing about this function is that it has an analyticalcontinuation for s = −1, whereζ (−1) = − 112 ,which is exactly what we need. Using this, we find that−1 = 1 ∞ 2 (D − 2) ∑nn=1= − 1 (D − 2)24⇓n=1D = 26. (2.126)Again, this is <strong>of</strong> course far from being a rigourous pro<strong>of</strong> that space-time should be 26-dimensional, but it does make it seem very plausible. Readers who demand a rigourouspro<strong>of</strong> are referred to e.g. §2.3.3 in [9], where the so-called no-ghost theorem gets proven.Still other ways exist, amongts which we note one that cleverly exploits the Virasoroalgebra, which can be found e.g. on p.44-46 in [2].At last, we are now in a position to analyse the spectrum <strong>of</strong> the bosonic string.

CHAPTER 2. BOSONIC STRINGS 34<strong>Open</strong> string spectrumAs always, we construct the spectrum <strong>of</strong> the theory by acting on the ground state withthe raising operators. Before doing this, we remind ourselves that the groundstate hadM 2 equal toα ′ M 2 |0〉 = −a|0〉 = −1|0〉. (2.127)Hence, we see that our groundstate is in fact a tachyon. This, to say the least, is badnews, but there is not much we can do about it. 9Next, we turn to the first excited state, generated by acting once upon the groundstatewith a raising operator,α i −1|0〉, (2.128)and as we saw earlier, this should be a massless particle. Since there are 24 raisingoperators, we find ourselves with 24 massless states.Going one step further, we find several ways to create the next state, namelyα−2|0〉 i or α−1α i j −1 |0〉. (2.129)Both ways give rise to a massive particle with α ′ M 2 = 1. The first possibility clearlyleads to 24 different states. Recalling that the “non-zero mode” operators commute, thesecond one amounts to24 2 − 242+ 24 =24 (24 − 1) + 482=24 · 252= 300 states, (2.130)so in total, we find ourselves with 324 states with α ′ M 2 = 1.To construct the rest <strong>of</strong> the spectrum, which is infinite, one just has to carry onthis game as long as one wishes. This is however a nice time to introduce a nice littletool that allows one to easily compute the number <strong>of</strong> states at each and every masslevel, which responds to the name <strong>of</strong> generating function. As an appetizer, consider theidentity1+∞∑1 − x = 1 + x + x2 + x 3 + · · · = x n . (2.131)n=0In this case, we see that expanding 11−xgenerates all the terms 1 + x + . . .. We can putthis knowledge to our advantage by considering how we construct open string states.The ground state is unique. Then we act on the ground state with raising operators, andwe have 24 sets <strong>of</strong> these. For the first excited level, we have 24 choices correspondingto the 24 α−1 i operators. Going one step further, we also need to consider the αi −2operators, even one step further also α−3 i and so on. But if we consider all the modesseparately for a second, and considering only one direction, say i = 1, we see that theα−1 1 operator will give rise to one level one state, one level two state ((α1 −1 )2 ), one levelthree state, etc. Assuming x represents a state, and that the exponent <strong>of</strong> x corresponds9 An interpretation for this has been found as the decay <strong>of</strong> D25-branes into closed strings, but we willcome back to this in due time.

CHAPTER 2. BOSONIC STRINGS 35to its mass level (0 = vacuum, . . .), we see that this is in fact encoded in Eq. 2.131. Allterms have coefficient one, expressing that we have exactly one state at each mass level.The mode two operator α−2 1 will give one level two state, one level four, etc. Thiscan be summarized into1+∞∑1 − x 2 = 1 + x2 + x 4 + x 6 + · · · = x 2n . (2.132)n=0To create states from the vacuum state by combining raising operators <strong>of</strong> differentmode number, we need to multiply them, e.g. α−1 1 α1 −2 |0〉. Accounting for the fact thatwe have an infinite set <strong>of</strong> them, this suggests the generalization <strong>of</strong> the above reasoningto( ) ( ) ( )1 1 1+∞ ∏ 1×1 − x 1 − x 2 ×1 − x 3 × · · · =1 − xn. (2.133)n=1Since we have one set <strong>of</strong> these operators in all 24 transverse directions, we should accountfor this by applying the straightforward modification( ) 1 24 ( ) 1 24 ( ) 1 24 +∞ ∏×1 − x 1 − x 2 ×1 − x 3 × · · · =n=11(1 − x n )24.(2.134)Expanding this (which takes some time doing it by hand, but lucky for us we havecomputers to do that stuff for us 10 ), we find+∞ ∏n=11(1 − x n ) 24 = 1 + 24x + 324x2 + 3200x 3 + . . . , (2.135)which indeed confirms our earlier result that we have one groundstate, 24 massless statesand 324 level two states, and tells us that we would have found 3200 level three statesif we would have gone on. Note that if we would prefer that the exponent <strong>of</strong> x wouldrepresent the mass <strong>of</strong> the state expressed in units <strong>of</strong> a, we could achieve this by upgradingour generating function toClosed string spectrum1+∞ ∏ 1x (1 − xn=1n ) 24 = 1x−1 + 24x 0 + 324x 1 + 3200x 2 + . . .. (2.136)The closed string has two sectors: left-movers and right-movers. The level-matchingcondition Eq. 2.102 however tells us we have to excite both sectors as many times.Other than that, the spectrum is obtained in exactly the same way, and we again finda tachyon in the ground state,α ′ M 2 |0〉 = −4|0〉. (2.137)10 In Mathematica, one can use the line “Series[Product[(1 − x n ) −24 , {n, 1, y}], {x, 0, z}]” and replacey and z by integers <strong>of</strong> choice with y ≥ z to obtain the right accuracy at level z in the expansion.

CHAPTER 2. BOSONIC STRINGS 36For the first excited level though, we now have many more possibilities, since thesestates are given byα−1˜α i j −1 |0〉, (2.138)and both excitations are essentialy independent. This results in 24 2 = 576 masslessstates. However, the fact that these excitations are indepent has a huge consequence,namely it allows us to decompose the states in three distinct parts as follows:[α−1α i j −1 |0〉 = α[i −1˜αj] −1 |0〉 + α −1˜αj) (i−1 − 1D − 2 δij α−1˜α k −1k]|0〉+ 1D − 2 δij α k −1˜α k −1|0〉, (2.139)where [...] stand for anti-symmetrization and (...) stand for symmetrization. Thus we obtainan anti-symmetric part, a traceless symmetric part, and a part that is proportionalto the unit matrix. And now comes the moment you have all been waiting for: thegraviton has entered the building! Admittedly, we are skipping a few steps here, butwe can nevertheless already identify the traceless symmetric part <strong>of</strong> Eq. 2.139 with amassless spin two particle described by a traceless symmetric rank two tensor, usuallydenoted g µν . It turns out though that if one would investigate how this particle couplesto matter and to itself, it does indeed behave like a graviton (cfr. [14], Chapter 15).Furthermore, we also find ourselves in the lovely company <strong>of</strong> a massless antisymmetricrank two tensor, which is called the Kalb-Ramond field and denoted B µν , and a masslessscalar which responds to the name dilaton field and denoted Φ, both <strong>of</strong> which will playvery important roles later on.Note that in the case <strong>of</strong> the closed string, the generating function trick describedfor the open string case does not apply. The origin for this lies in the level matchingcondition, Eq. 2.102, combined with the commutators for the modes, Eq. 2.53. The commutationrelations show that an m-level operator weights more than an (n < m)-leveloperator, resulting in the fact that one cannot simply write down an elegant generatingfunction for this situation, but has to verify the level-matching on a level-per-level basis.2.7 Oriented vs. unoriented stringsWe note in passing that one could define a projection operation that projectsσ −→ π − σ, (2.140)i.e. it flips all points on the string around the centerpoint. The consequence <strong>of</strong> thisoperation for the mode expansions isopen strings: α µ m ←→ (−1) m α µ m, (2.141)closed strings: α µ m ←→ ˜α µ m. (2.142)<strong>String</strong>s unaffected by this operation are called unoriented, in contrast to their brethrenwho do feel the operation and are called oriented.

CHAPTER 2. BOSONIC STRINGS 37The first relation amounts to saying that the spectrum <strong>of</strong> the unoriented open stringcontains only states with even mode number. In particular, it contains no photon(represented by the massless states).An important consequence <strong>of</strong> the second relation is that as far as the spectrum <strong>of</strong>the unoriented closed bosonic string is concerned, only states symmetric in α and ˜αsurvive, implying that the antisymmetric B µν tensor does not. Giving away a spoiler bysaying that strings can be shown to acquire charge under the B µν field, this and previousparagraphs show that unoriented theories are vastly different from oriented ones.In this document, we will only consider oriented theories.2.8 SummaryIn this chapter, we started out by considering the simplest possible bosonic string action,namely the Nambu-Goto action. We noted that it was difficult to quantize, and sointroduced a world sheet metric so as to be able to switch to the Polyakov action. Wethen analysed the symmetries <strong>of</strong> the Polyakov action, which allowed us to choose agauge in which deriving the equations <strong>of</strong> motion <strong>of</strong> the bosonic fields was particularlyeasy. Demanding that the boundary term <strong>of</strong> the varied action vanish, we found thatit was possible to impose several boundary conditions on our string, which gave birthto different types <strong>of</strong> strings. By their nature, these strings, and more specifically thesolutions to their equations <strong>of</strong> motion, allowed to be Fourier decomposed. This led to acanonical quantization in which the classical modes where promoted to quantum raisingand lowering operators. However, there was an ordering ambuigity to be solved, andthis appeared to be simpler to do in the light-cone gauge. Switching to this gauge alsoallowed us to derive the number <strong>of</strong> space-time dimensions in order for the theory to beLorentz invariant, and finally, to easily construct the spectrum <strong>of</strong> the bosonic string(s).All this been said, what we have seen in this chapter is only a minuscule part <strong>of</strong> whatbosonic string theory is comprised <strong>of</strong>. One could write an entire book about the bosonicstring alone, an endeavour which has been succesfully accomplished most notably byPolchinski [18].

Chapter 3Superstrings“One <strong>of</strong> the problems has to do with the speed <strong>of</strong> lightand the difficulties involved in trying to exceed it. Youcan’t. Nothing travels faster than the speed <strong>of</strong> light withthe possible exception <strong>of</strong> bad news, which obeys it ownspecial laws. The Hingefreel people <strong>of</strong> Arkinto<strong>of</strong>le Minordid try to build spaceships that were powered by badnews but they didn’t work particularly well and were soextremely unwelcome whenever they arrived anywherethat there wasn’t really any point in being there.”Douglas AdamsHitchhiker’s Guide to the GalaxyThe bosonic string is a versatile object, but as things usually go with prototypes, it isnot exactly there yet. It mainly has two flaws. First, its spectrum always contains atachyon, something we do not exactly appreciate (Adams seems to rightly suggest it isthe quantum <strong>of</strong> bad news). Second, it is purely bosonic, meaning it can not possiblydescribe fermions. Being that fermions in large part make up the world as we know it,this is <strong>of</strong> course a serious issue.So the question now becomes how we can possibly alter bosonic string theory suchthat it also incorporates fermions. The answer turns out to be to introduce a new symmetrycalled supersymmetry, that relates, at least on some level, bosons and fermions.Using this symmetry, one can expand the bosonic string action with an extra fermionicterm, thus creating what is called the superstring. As in the case <strong>of</strong> the bosonic string,the equations <strong>of</strong> motion <strong>of</strong> this superstring will lead to several possible boundary conditions,resulting in different types <strong>of</strong> superstrings. Possibilities to combine boundaryconditions for open strings will however increase, and have considerable consequencesfor the spectrum <strong>of</strong> said strings, resulting in different types, or flavours, <strong>of</strong> superstringtheories.38

CHAPTER 3. SUPERSTRINGS 39First <strong>of</strong> all, it should be pointed out that two main methods exist to introducesupersymmetry to the bosonic string theory. The first method is called the “Ramond-Neveu-Schwarz” (RNS) method, and starts from a position <strong>of</strong> making the world sheetsupersymmetric. The second method is called the “Green-Schwarz” (GS) method, andstarts from a position <strong>of</strong> making space-time supersymmetric. Both methods have theiradvantages and disadvantages, however the second one has the drawback <strong>of</strong> being vastlymore technical and hard to quantize. Therefore, we will disregard it completely in thisdocument, and will concentrate instead on the RNS formalism. Both formalisms areequivalent for a 10-dimensional space-time, which turns out to be the critical dimensionfor superstring theory. Parties interested in the GS formalism are referred to e.g.Chapter 5 in both [2] and [9].We will not be concerned with specific details and applications, but rather focus onpresenting the method that allows one to write a SUSY string action, what this entailsfor the spectrum <strong>of</strong> said strings, and how this results in different superstring theories.3.1 Superstring actionBut let us start at the beginning. If we want to include fermions in our theory, the firstthing we need is something that behaves like a fermion. To this end, we introduce fieldsψ α µ with µ ∈ {0, . . .,D − 1} and α ∈ {0, 1}, which are physical objects that behave liketwo-component spinors on the world sheet (hence their spinor index α), and as vectorsin D-dimensional space-time (hence their Lorentz index µ). Note that we introduceobjects that behave as fermions on the world sheet. We do not yet have objects thatbehave as fermions in spacetime, but we will come to that soon enough.If we want to add these new fields to our existing bosonic string action, options arelimited as to what we can do. Since we are assuming a supersymmetric relation betweenthe bosonic fields X µ and ψ µ , give by the simultaneous transformations{δX µ = ¯ǫψ µ ,δψ µ = ρ α ∂ α X µ (3.1)ǫ,in which ǫ represents an infinitesimal parameter <strong>of</strong> this supersymmetry transformation,we should <strong>of</strong> course add as many fermionic fields as we had bosonic fields, meaning onefor each space-time dimension. The form <strong>of</strong> the action that imposes itself is that <strong>of</strong> astandard Dirac action for massless fermions, hence the superstring action readsS = − 1 ∫2πα ′ d 2 σ ( ∂ α X µ ∂ α X µ + ¯ψ µ ρ α )∂ α ψ µ . (3.2)This action is symmetric under Eq. 3.1, and in it, ρ α represents the two-dimensionalDirac matrices, obeying the Clifford algebra 1 defined by{ρ α , ρ β} = 2η αβ½2×2. (3.3)1 See e.g. Chapter 3 in [22].

CHAPTER 3. SUPERSTRINGS 40In D dimensions, and for D even, there are 2 ⌊D/2⌋ <strong>of</strong> them. 2 In two dimensions, one canchoose a real, or Majorana, representation, and we choose[ ][ ]ρ 0 0 −1=; ρ 1 0 1= . (3.4)1 01 0The spinors ψ µ also are real, and hence have two (real) components, which, in anticipation<strong>of</strong> the use <strong>of</strong> light-cone coordinates, we denote[ψ µ ψµ]−=ψ µ . (3.5)+We further define¯ψ = ψ † β ; β = iρ 0 . (3.6)Note that since ψ µ is real, ψ † = ψ T . It should further be pointed out that the fields ψ µanticommute, as they are expected to, so{ψ µ , ψ ν } = 0. (3.7)This is only true in the classical theory though. Although we will not go into this, wemention that quantum mechanically this relation gets upgraded to{ψµA (σ, ( τ),ψν B σ ′ , τ )} = πη µν δ AB δ ( σ − σ ′) . (3.8)We see that at first sight, we are confronted with the same ghost problem we had for thebosonic string. In order to eliminate these, one has to introduce supersymmetry, whichresults in an extension <strong>of</strong> the Virasoro algebra (called the super-Virasoro algebra), inturn leading to altered super-Virasoro constraint equations, which will do the trick.For the sake <strong>of</strong> completeness, we mention that while we introduced, or even imposed,supersymmetry by hand, one could also define a superspace, which amounts to addinganticommuting coordinates to an arbitrary space-time. The advantage <strong>of</strong> this is that inthis formulation supersymmetry if manifest. For our purposes however, this does notmake any difference for the results we want to obtain, and would only needlessly makethings more complicated. Interested parties are referred to e.g. [9], §4.1.2.3.2 Equations <strong>of</strong> motionWe first <strong>of</strong> all note that the bosonic term in the string action is unaffected by thepresence <strong>of</strong> the fermionic term. Hence, nothing will change in the equations <strong>of</strong> motion,and subsequent spectrum <strong>of</strong> the bosonic part <strong>of</strong> the string. This allows us to single outthe fermionic part <strong>of</strong> the equation, and consider it separately. Let us denote this partbyS F = − 1 ∫2πα ′ d 2 σ ( ¯ψµ ρ α )∂ α ψ µ . (3.9)2 For D uneven there is one more, but we will not be confronted with this possibility.

CHAPTER 3. SUPERSTRINGS 41Using Eqs. 3.4, 3.5 and 3.6, and dropping the Lorentz index for convenience, we findthat(¯ψρ α ∂ α ψ = i [ψ − ψ + ] ( ρ 0) [ ]2 ∂τ ψ −+ [ψ∂ τ ψ − ψ + ] ( ρ 0 · ρ 1) [ ])∂ σ ψ −,+ ∂ σ ψ +( [ ] [ ])−∂τ − ∂= i [ψ − ψ + ] σ 0 ψ−,0 −∂ τ − ∂ σ ψ += −i (ψ − (∂ τ + ∂ σ ) ψ − + ψ + (∂ τ − ∂ σ ) ψ + ),= −2i (ψ − ∂ + ψ − + ψ + ∂ − ψ + ), (3.10)where in the last line we have used Eqs. 2.22 and 2.23. Reinsertion <strong>of</strong> this result in Eq.3.9 leaves us withS F = i ∫πα ′ d 2 σ (ψ − ∂ + ψ − + ψ + ∂ − ψ + ) , (3.11)which lends itself nicely to deriving the equations <strong>of</strong> motion <strong>of</strong> ψ + and ψ − . Varying thisaction, we find thatδS F = i ∫ ∫ ππα ′ dτ dσ [(δψ − ) ∂ + ψ − + ψ − ∂ + (δψ − ) + (δψ + ) ∂ − ψ + + ψ + ∂ − (δψ + )] ,0= i ∫ ∫ ππα ′ dτ dσ[−2 {(∂ + ψ − ) δψ − + (∂ − ψ + ) δψ + }0+ {∂ + (ψ − δψ − ) + ∂ − (ψ + δψ + )}], (3.12)in which we have used the anticommutativity <strong>of</strong> the spinors, and from which we immediatlysee that∂ + ψ − = 0 = ∂ − ψ + , (3.13)which are the equations <strong>of</strong> motion for the fermionic fields.3.3 <strong>Bound</strong>ary conditions and expansionsEq. 3.12 still leaves us with boundary terms, namelyδS F = i ∫ ∫ ππα ′ dτ dσ[∂ τ {(ψ − δψ − ) + (ψ + δψ + )}0+ ∂ σ {(ψ − δψ − ) − (ψ + δψ + )}]. (3.14)We disregard the ∂ τ term as usual, which leaves onlyδS F = i ∫πα ′ dτ {[(ψ − δψ − ) − (ψ + δψ + )] σ=π− [(ψ − δψ − ) − (ψ + δψ + )] σ=0} . (3.15)This should <strong>of</strong> course vanish, and this can be achieved, as in the bosonic case, byimposing different conditions, leading to two different “sectors.” Different combinations<strong>of</strong> these sectors will eventually lead to different types <strong>of</strong> superstrings, open as well asclosed.

CHAPTER 3. SUPERSTRINGS 423.3.1 Closed stringsWe start by considering the possibility thatψ + (σ, τ) = ±ψ + (σ + π, τ), (3.16)ψ − (σ, τ) = ±ψ − (σ + π, τ). (3.17)Note that these suppositions imply that the same relations also apply to the variationsδψ ± . Furthermore, we see that if the + sign is chosen, the functions are periodic andare said to obey Ramond boundary conditions (R), if − is chosen they are antiperiodicand obey Neveu-Schwarz boundary conditions (NS). ψ + corresponds to left-movers, ψ −to right-movers, and the periodicity condition for them can be chosen independantly,resulting in four different possible combinations, namely, choosing to write them as(left-movers,right-movers), (NS,NS), (NS,R), (R,NS) and (R,R).To realize these different boundary conditions, the following expansions can be applied:3 R left-movers: ψ + (σ, τ) = ∑ m∈˜dµ m e −2im(τ+σ) , (3.18)R right-movers: ψ − (σ, τ) = ∑ n∈d µ ne −2in(τ−σ) , (3.19)NS left-movers:ψ + (σ, τ) = ∑˜bµ r e −2ir(τ+σ) , (3.20)NS right-movers:r∈+ 1 2ψ − (σ, τ) = ∑b µ se −2is(τ−σ) . (3.21)s∈+ 12We see that the (anti)periodicity is reflected in the (half-)integer summing indices. Fromhereon, we will follow the convention <strong>of</strong> using indices (m, n) for integer numbers, and(r, s) for half-integer numbers.3.3.2 <strong>Open</strong> stringsWhen considering open strings, the names Ramond and Neveu-Schwarz boundary conditionsare still used, but they correspond to different conditions. Nevertheless, they arevery similar, explaining the double usage.Since now the positions (σ = 0) and (σ = π) correspond to two different and a prioriunrelated positions on the string, the two terms in Eq. 3.15 should vanish seperately.Hence, we see that{ψ − (0, τ)δψ − (0, τ) − ψ + (0, τ)δψ + (0, τ) = 0,(3.22)ψ − (π, τ) δψ − (π, τ) − ψ + (π, τ)δψ + (π, τ) = 0.3 Almost every reference uses b and d to denote the NS and R modes respectively, except for [14] whouse b for both, and differentiate only by means <strong>of</strong> their indices (m, n) or (r, s).

CHAPTER 3. SUPERSTRINGS 43These equations can be resolved by relating the boundary conditions for ψ − and ψ + ,which in turn impose the same conditions for their respective variations, in the followingway: {ψ − (σ ⋆ , τ) = ±ψ + (σ ⋆ , τ),δψ − (σ ⋆ , τ) = ±δψ + (σ ⋆ (3.23), τ),for σ ⋆ ∈ {0, π}. Due to the fact that ψ ± and their variations appear quadratically inthe boundary conditions, their absolute sign is <strong>of</strong> no importance. What matters is therelative sign between σ = 0 and σ = π. Therefore, it is conventionally chosen to setNow, our two sectors appear. The conditionψ − (σ = 0) = ψ + (σ = 0) . (3.24)ψ − (σ = π) = +ψ + (σ = π) (3.25)corresponds to the open string Ramond boundary condition, the caseψ − (σ = π) = −ψ + (σ = π) (3.26)is the open string Neveu-Schwarz boundary condition. The expansions are similar tothose for the closed string, but containing only one set <strong>of</strong> expansion coefficients.⎧⎪⎨ψ + (σ, τ) = 1 ∑√2Ramond open string:m∈d µ me −im(τ+σ) ,⎪⎩ψ − (σ, τ) = √ 1 ∑(3.27)2n∈d µ ne −in(τ−σ) ,Neveu-Schwarz open string:⎧ψ + (σ, τ)⎪⎨= 1 ∑√2ψ − (σ, τ)⎪⎩= √ 1 ∑2r∈+ 1 2s∈+ 1 2b µ re −ir(τ+σ) ,b µ se −is(τ−σ) .(3.28)The condition Eq. 3.24 is trivially satisfied. Further remark that for the NS open string,ψ µ 1 ∑− (π, τ) = √2r∈+ 1 2b µ re −ir(τ) e irπ = −ψ µ + (π, τ),as expected, since e irπ = −e −irπ for r ∈+ 1 2. For the R open string, given thate inπ = e −inπ with n ∈, we also verify that the corresponding boundary condition isindeed satisfied.

CHAPTER 3. SUPERSTRINGS 443.4 Quantization and spectrumTurning to the quantization <strong>of</strong> these strings, we will again start by promoting the expansionmodes to quantum operators using the canonical formalism, switching to (spacetime)light-cone gauge in order to construct the spectra.First thing to note is that the behaviour <strong>of</strong> the bosonic degrees <strong>of</strong> freedom has notchanged, and as a consequence, also the quantization <strong>of</strong> their modes remains the same,i.e.[α µ m, α ν n] = [˜α µ m, ˜α ν n] = mδ m+n,0 η µν ; [α µ m, ˜α ν n] = 0. (3.29)We could have instead chosen to use a instead <strong>of</strong> α, in which case the above relationsbecome[a µ m, a ν n] = [ã µ m, ã ν n] = δ m+n,0 η µν ; [a µ m, ã ν n] = 0. (3.30)The behaviour <strong>of</strong> the fermionic modes is very similar, the biggest difference being thatthey obey anticommutation relations instead <strong>of</strong> commutation relations, as a consequence<strong>of</strong> them being fermionic. As such, the quantization prescription becomes{. . .,...} −→ −i {. . .,...} , (3.31)where the left anticommutator represents the classical one, and the one on the right isthe quantum version. We also note that the fact that ψ µ are Majorana spinors getsinherited by the modes in that sense that they need to satisfy the condition(b µ r) † = b µ −r ; (d µ r) † = d µ −r , (3.32)and analogous for their tilded brethren. The result <strong>of</strong> all this are the following relations:}{ }{b µ r,b ν s} ={˜bµ r ,˜b ν s = η µν δ r+s ; b µ r,˜b ν s = 0, (3.33){{d µ m, d ν n} = ˜dµ m , ˜d}{νn = η µν δ m+n ; d µ m, ˜d}νn = 0. (3.34)Notice that the b µ and ˜b µ operators do not contain a zero mode.Before moving on to the spectrum, and although we will not go into the details, wemention that also in the superstring case one can expand the energy-momentum tensor,as well as the supercurrent coming from the global world sheet supersymmetry. Themodes <strong>of</strong> these expansions form the super-Virasoro algebra. Furthermore, also analogousto the bosonic string case, upon normal ordering <strong>of</strong> the modes one induces two orderingcontants a R and a NS , respectively for the Ramond and Neveu-Schwarz sectors. Byplaying a clever game involving the super-Virasoro algebra and physical states, one canderive the results that a R = 0, a NS = 1 2and D = 10 for superstrings. The same resultcould be obtained by first constructing the spectrum <strong>of</strong> the strings in the space-timelight-cone gauge, and then applying a zeta-regularization technique similar to the onewe saw in Chapter 2 for the bosonic string. We will consider these results as known,and immediatly use them in what follows.It is worth pointing out that Eq. 3.2 still describes one, and only one, string. Wedo not get two strings: one which behaves like a boson, and one which behaves like a

CHAPTER 3. SUPERSTRINGS 45fermion. We still have only one string, only this time around this string has bosonic andfermionic degrees <strong>of</strong> freedom on the world sheet. This means that general states <strong>of</strong> thissingle string will be obtained by acting with both bosonic and fermionic raising operatorson the vacuum. Hence, when we state that a R = 0 and a NS = 1 2, we in fact state thatthe normal ordering constant that results from the sum <strong>of</strong> the bosonic and fermionicdegrees <strong>of</strong> freedom equals 0 when considering R modes, and 1 2when considering NSmodes.On a sidenote, remark that the fact that a R = 0 can be intuitively understoodby realizing that both α and d are integer moded operators, but that since one obeyscommutation relations and the other anticommutation relations, their normal orderingcontants will carry opposite signs, and hence annihilate.Moving on to the construction <strong>of</strong> the spectra, we will now switch to space-time lightconegauge. This is possible because <strong>of</strong> residual gauge symmetries left after gauge fixingthe world sheet metric, similar to the bosonic string case. Moreover, a similar thing alsoapplies to the fermionic maps, as a result <strong>of</strong> which we can setX + (σ, τ) = x + + p + τ, (3.35)ψ + (σ, τ) = 0. (3.36)If one were to go one step further, one would find that one could still relate the X −and ψ − modes to the modes <strong>of</strong> the transverse oscillators, thereby effectively eliminatingtwo sets <strong>of</strong> modes, and keeping only the transverse (raising and lowering) modes. Thezero modes are still present for all directions, where applicable. As for the bosonic case,transverse modes will be denoted with a superscript i . Using the previously stated factthat D = 10 for superstring theories, we are thus left with eight transverse directions.3.4.1 <strong>Open</strong> stringsSpecializing to the case <strong>of</strong> open superstrings, we need to differentiate between the Ramondand Neveu-Schwarz sectors.Neveu-Schwarz sectorApplying the mass-shell condition to the NS sector results in the mass formulaα ′ M 2 =+∞∑n=1α i −nα i n ++∞∑r= 1 2rb i −rb i r − 1 2 , (3.37)where summation over i is understood, and where as anticipated earlier we alreadyincorporated the result a NS = 1 2. This immediatly tells us thatα ′ M 2 |0〉 NS = − 1 2 |0〉 NS, (3.38)where |0〉 NS describes the vacuum <strong>of</strong> the open string NS sector. We find our old foo,the tachyon, back with a vengeance. It appears that simply adding fermionic degrees <strong>of</strong>

CHAPTER 3. SUPERSTRINGS 46freedom to our theory does not suffice to get rid <strong>of</strong> it. We will need to be more clever ifwe want to beat it. Remark that this ground state is unique, hence describes a spin 0state, hence is bosonic.We can construct a first excited state by acting once with a b raising operator.Remember that the b’s are half-integer moded, and thus b i is lower in energy than− 1 2α−1 i . Thus, we identify the first excited state asb i |0〉− 1 NS . (3.39)2We have eight sets <strong>of</strong> transverse oscillators, and thus also eight b i operators. So we find− 1 2ourselves with a state that transforms under SO(8), and hence is a space-time vectorboson. This should be a massless state (recall the similar discussion in Chapter 2). Asecond look at Eq. 3.37 learns us that acting with α−n i increases the mass with m, 4while acting with b i −r increases the mass with r. Since our groundstate has mass − 1 2 ,and r = 1 2for our first excited state, we indeed verify that this is true. (Not knowing thevalue <strong>of</strong> a NS , and applying the inverse reasoning is again a way to see that a NS shouldbe 1 2.) This is good because we find back our beloved photon.The second excited state can be obtained either by applying once an α raising operator,or by applying two b raising operators. Beware though, the b’s satisfy anticommutationrelations, and so we can not excite twice in the same direction with thesame operator, as ( b i 2r)= 0. This greatly diminishes the number <strong>of</strong> b raising operatorcombinations we can form. In this particular case, we can choose one out <strong>of</strong> eight b i − 1 2operators for the first operator, and one <strong>of</strong> the seven remaining b j for the second one.− 1 2Also keeping in mind that b i b j = −b j b i , further diminishing the number <strong>of</strong> possibilitieswith a factor two. We are left with (8 ×7)/2 = 28 distinct possible states. Add− 1 2 − 1 − 1 − 1 2 2 2to these the eight states we obtain by applying an α− i operator, and we obtain 36 statesin total for the second excited state with mass α ′ M 2 = 1 2 .Generalizing this to the rest <strong>of</strong> the spectrum, we find that we can encode the number<strong>of</strong> states at every mass level using the generating functionf ′ NS (x) =+∞ ∏n=1(1 + x n−1 21 − x n ) 8. (3.40)We recognize the denominator as encoding states resulting from acting with bosonicmodes. The numerator encodes the states resulting from acting with fermionic modes.( 8Indeed, 1 + x 2) n−1 is in fact nothing but a polynomial <strong>of</strong> which the coefficients willequal ( 8m), with m ∈ {1, . . .,8}, corresponding exactly to all possible ways <strong>of</strong> combiningeight anticommuting b operators with mode number n − 1 2 . Defining4 See Eq. 3.29.f NS (x) = 1 √ xf ′ NS (x) = 1 √ x +∞ ∏n=1(1 + x n−1 21 − x n ) 8, (3.41)

CHAPTER 3. SUPERSTRINGS 47to again make the exponent <strong>of</strong> x match the mass <strong>of</strong> the appropriate state, the first fewterms in the expansion <strong>of</strong> f NS readRamond sectorf NS (x) = 1x −1 2 + 8x 0 + 36x 1 2 + 128x 1 + 402x 3 2 + 1152x 2 + . . .. (3.42)The mass-shell condition applied to the Ramond sector delivers us the mass formulaα ′ M 2 =+∞∑n=1α i −nα i n ++∞∑m=1d i −md i m, (3.43)where again we incorporated the condition a R = 0. Applying α i −n or d i −m thus raisesthe mass with n or m units respectively.The good news is that we see that there is no tachyon in the open string R sector!The, at first sight, bad news is that the groundstate is degenerate. However, we will beable to turn this to our advantage later on. Recall that the d modes satisfy the algebra{d µ m, d ν n} = η µν δ m+n,0 .The big difference with the NS case is that this time around there are anticommutingzero modes which, as the above equation demonstrates, satisfy a ten-dimensional Cliffordalgebra 5 (except for a factor <strong>of</strong> 2, cfr. Eq. 3.3, which can be accounted for by rescalingthe d 0 operators with a factor √ 2.), indeed confirming that the groundstate is degenerate.Given that the d µ ’s satisfy a Clifford algebra, the groundstates in the R sector shouldform a representation <strong>of</strong> this algebra, and hence we can write thatd µ 0 |a〉 = √ 1 Γ µ2ba|b〉, (3.44)with (|a〉, |b〉) groundstates and Γ µ ba forming a basis <strong>of</strong> the 2⌊D/2⌋ -dimensional Cliffordalgebra. Given that D = 10 in this case, this is a 32-dimensional algebra. It can beshown that these ten-dimensional spinors can actually be reduced to eight-dimensionalMajorana-Weyl spinors having 16 real components, amounting to two times eight groundstates <strong>of</strong> two opposite chiralities; sixteen in total if you prefer. We denote these by |a〉and |ā〉. For a further discussion on how one can combine the different d µ 0 operators into“raising” and “lowering” operators, see §3.1 in [17].The fact that the groundstate is degenerate immediatly makes the number <strong>of</strong> statesin the R sector grow considerably bigger compared to the NS sector. On the other hand,the NS sector contains twice as many mass levels, because the b operators are half-integermoded. This already points out two discrepancies between both spectra: a difference inmass levels, and a difference in states in corresponding mass levels. Noticing that theR ground states are space-time spinors, we can identify them as space-time fermions.Acting on them with the raising operators keeps creating space-time spinor states, and5 Remember that switching to space-time light-cone gauge did not eliminate the zero modes.

CHAPTER 3. SUPERSTRINGS 48hence, fermions. Hence we see that if we keep all states <strong>of</strong> both sectors, we will obtain adifferent number <strong>of</strong> space-time fermions and bosons, while if space-time supersymmetrywere respected these numbers should be equal. Lucky for us, there is a way to obtainsuch a space-time supersymmetric spectrum, and we will soon enough see how. Butfirst, we finish contructing the unconstrainted R spectrum.Apart from the degeneracy in ground states, the spectrum is constructed in the sameway as always. Only this time, we can let the raising operators act on several groundstates. We thus find that the first excited level with mass α ′ M 2 = 1 can be reached byapplyingα i −1|a〉 ; α i −1|ā〉 ; b i −1|a〉 ; b i −1|ā〉. (3.45)We thus find that there are 8 × 16 × 2 = 256 states at the first mass level, 128 for eachchirality. The generating function reads+∞ ∏f R (x) = 16The first few terms in the expansion aren=1( 1 + xn1 − x n ) 8. (3.46)f R (x) = 16x 0 + 256x 1 + 2304x 2 + 15360x 3 + . . .. (3.47)Notice by comparing this to Eq. 3.42 that at corresponding mass levels, the Ramondsector contains twice as many states.3.4.2 GSO projectionAs we have just seen, the NS sector contains a tachyon, which is something we wouldvery much like to get rid <strong>of</strong>. Not only that, but moreover there is a different number <strong>of</strong>states in corresponding mass levels between the NS sector (which are space-time bosons)and R sector (which are space-time fermions). This is bad, because if supersymmetrywere to be preserved, both numbers would be equal.In order to solve this sneaky problem, we will use an ingenious little sorting device,called the GSO projection. 6 Looking back at Eqs. 3.42 and 3.47, one sees that it wouldturn out really great if one were to find a way to eliminate the half-integer mass levels<strong>of</strong> the NS sector, and half the states <strong>of</strong> the R sector. This second part does not seemtoo complicated; simply project out one <strong>of</strong> both chiralities. For the first requirement,we will need to find a way to separate half-integer from integer mass modes.To reach our objectives, we introduce two operators, one for the NS sector, and onefor the R sector, but both responding to the name <strong>of</strong> G-parity operator. For the NSsector, we define this operator as 7G = (−1) F+1 with F =∞∑b i −rb i r. (3.48)r= 1 26 Originally introduced by Gliozzi, Olive and Scherk.7 Some references include the “-1” term in the definition <strong>of</strong> F.

CHAPTER 3. SUPERSTRINGS 49F, when acting on a NS state, returns the number <strong>of</strong> times a fermionic operator hasbeen acted on the groundstate in order to obtain this state. So we see thatG|0〉 = −1, (3.49)and that acting on other states, G will be 1 for half-integer mass levels (levels, notmasses!) and -1 for integer mass levels. The GSO projection for the NS sector consistsin keeping only those states with positive G-parity. This way, we eliminate the tachyonfrom the spectrum, as well as all other states with half-integer mass, which was exactlywhat we wanted.For the R sector, G becomes a generalized chirality operator,∞∑G = Γ 11 (−1) F with F = d i −rd i r, (3.50)with Γ 11 the ten-dimensional chirality operator. The GSO projection in the R sectorconsists <strong>of</strong> keeping only one <strong>of</strong> both chiralities. Which one is purely a matter <strong>of</strong> choice.This effectively eliminates half <strong>of</strong> all states at every mass level.Since the result is too beautiful not to mention, we will take a quick look at themodified generating functions for the truncated (i.e. what is left after performing theGSO projection) spectra. To update f R (x) (Eq. 3.46), it suffices to change the factorin front <strong>of</strong> the product from 16 to 8:+∞ ∏f R (x) −→ fR GSO (x) = 8n=1n=1( 1 + xn1 − x n ) 8. (3.51)In the NS sector, we need to subtract all states obtained by acting with an even number<strong>of</strong> raising b operators from the full spectrum. This can be achieved by considering thegenerating function1+∞ ∏√ xn=1(1 − x n−1 21 − x n ) 8= 1x −1 2 − 8x 0 + 36x 1 2 − 128x 1 + . . . , (3.52)in which we see that states corresponding to an odd number <strong>of</strong> b operators carry a − sign,while states corresponding to an even number <strong>of</strong> b operators carry a + sign. Howeverwe see that when we subtract Eq. 3.52 from the unconstrained spectrum generated byEq. 3.41, we will double the number <strong>of</strong> states that survive, so we need to divide by anadditional factor <strong>of</strong> two to rectify this. Concretely, our new generating function becomesf NS (x) −→ f GSONS (x) = 12 √ x⎡∞∏⎣n=1(1 + x n−1 21 − x n ) 8−∞∏n=1( ) ⎤ 81 − x n−1 2⎦.1 − x n (3.53)At first sight, this looks like bad news, but incredibly enough, it was proven by CarlGustav Jacobi in 1829 (!) that indeed,⎡ ( ) 8 ( ) ⎤ 81∞∏2 √ ⎣1 + x n−1 2∞∏ 1 − x n−1 2∞∏( )x 1 − x n −⎦ 1 + xn 81 − x n = 81 − x n , (3.54)n=1n=1n=1

CHAPTER 3. SUPERSTRINGS 50miraculously confirming our hopes that the NS and R sectors, after truncation, do indeedgive rise to an equal number <strong>of</strong> space-time bosons and fermions at each and every masslevel.Note that the match between both spectra does not prove space-time supersymmetry.But it is by any means a solid indication. Space-time supersymmetry can however beproven in the RNS formalism, but this is a highly non-trivial endeavour.3.4.3 Closed stringsAs we already saw earlier, for closed strings we can combine R and NS conditions infour distinct ways: (NS,NS), (NS,R), (R,NS) and (R,R). In order to obtain a space-timesupersymmetric spectrum though, we again need to perform the GSO projection. Forthe NS sector, this does not change much as far as our choices are concerned, howeverfor the R sector, this means we can choose between two chiralities, further increasingour number <strong>of</strong> possible choices.Essentially, this gives rise to two types <strong>of</strong> superstring theories: Type IIA and TypeIIB. Note that also in the superstring case, closed strings should satisfy the levelmatchingcondition, which still states that the left- and right-moving states shouldhave the same mass. From this we conclude that the full spectrum <strong>of</strong> both theories isobtained by tensoring left- and right-moving states <strong>of</strong> equal mass level.Type IIAThe Type IIA theory is the one that results when one chooses opposite chiralities for theleft- and right-moving R sectors. They should be chosen opposite, but which is whichdoes not matter. The massless states <strong>of</strong> this theory are summarized in Table 3.1. Thechiralities are denoted by R and ¯R. Only massless states are shown because first <strong>of</strong> allwe will only be interested in those, and second, the number <strong>of</strong> states at the first massivelevel is. . . massive.Looking at the massless states, our attention is grabbed by the (NS,NS) sector.Massless states in this sector are created by acting upon the vacuum state much in thesame way as for obtaining the massless bosonic closed string states (cfr. Eq. 2.138).This again suggests a decomposition in a tracless symmetric and antisymmetric part,and a part that is proportional to the unit matrix. In other words, we find back ourg µν , B µν and Φ fields!The (R,NS) and (NS,¯R) sectors give rise to two gravitinos with opposite chirality(one for each sector), and two dilatinos, also with opposite chirality. The fact that twogravitinos are present indicates that space-time is N = 2 supersymmetric. 8The (R,¯R) sector gives rise to 64 massless states (8×8). These can be decomposed ina vector gauge field A µ and an antisymmetric three-form gauge field A µνρ . Notice thatA µ contains 8 states, and that A µνρ contains ( 83)= 56 states, the sum <strong>of</strong> which indeed8 This means that there are two supersymmetric transformations, and hence the doublets (X, ψ) getexpanded to quadruplets (X 1, X 2, ψ 1, ψ 2).

CHAPTER 3. SUPERSTRINGS 51gives back our 64 massless states. These gauge fields are <strong>of</strong>ten called Ramond-Ramondfields for obvious reasons.Type IIBThe type IIB theory results from choosing identical chiralities for the R sector <strong>of</strong> bothleft- and right-movers. The massless states <strong>of</strong> this theory can also be found in 3.1. Weonly consider one chirality, namely R.The content <strong>of</strong> the (NS,NS) sector is exactly the same as for the Type IIA theory,and so also here we find back our g µν , B µν and Φ fields.The content <strong>of</strong> the (R,NS) and (NS,R) sectors is also the same, except for the factthat in the Type IIB theory the dilatinos and gravitinos have the same chirality.The (R,R) sector gives rise to quite different Ramond-Ramond gauge fields though,given that this time both chiralities are the same. It turns out that the massless TypeIIB (R,R) states can be regrouped in a zero-form scalar gauge field A, a two-formgauge(field A µν and a self-dual four-form gauge field A µνρσ . Note that also this time,8) (0 + 8)2 +1 82(4)= 1 + 28 + 35 = 64, as it <strong>of</strong> course should.Type IIAType IIB|0〉 RL × |0〉 ¯R |0〉 RL × |0〉 RR˜bi |0〉− 1 NSL × b j |0〉2− 1 NSR˜bi |0〉− 1 NSL × b j |0〉22− 1 NSR2˜bi |0〉− 1 NSL × |0〉 RR˜bi |0〉− 1 NSL × |0〉 RR22|0〉 ¯RL × b i |0〉− 1 NSR |0〉 RL × b i |0〉− 1 NSR22Table 3.1: Massless states for both Type II theories. The subscript L denotes left-movers,R stands for right-movers.Other superstring theoriesThe laws <strong>of</strong> logic dictate that if there are two Type II theories, there must be at leastone Type I theory. There indeed is such a theory, and it is a theory <strong>of</strong> unorientedstrings (remember: no B µν field, or any other antisymmetric state). For the sake <strong>of</strong>completeness we also mention the two heterotic string theories: E 8 × E 8 and SO(32).These are theories in which one <strong>of</strong> the left- and right-moving modes <strong>of</strong> the closed stringis supersymmetric, and the other one is bosonic. The names <strong>of</strong> these theories correspondto their symmetry groups.3.5 SummaryThis chapter has given a very brief introduction to the superstring, focusing mainlyon the construction <strong>of</strong> the open string spectrum and how to get rid <strong>of</strong> the problems

CHAPTER 3. SUPERSTRINGS 52associated with the unconstrained theory using the GSO projection, and on the twoType II theories resulting from the closed string spectra. The main purpose <strong>of</strong> thischapter was to obtain the massless fields contained in the Type II theories, given thatthese are the ones that interest us in the remainder <strong>of</strong> this writing, and getting there asfast as possible without being too sloppy. We saw that in the end we obtained spacetimebosons and fermions, and that their numbers agree at every mass level which is agood indication that space-time preserves supersymmetry. We also saw that both TypeII theories contain the (more or less) same g µν , B µν and Φ fields we already encounteredearlier in bosonic string theory. The Ramond-Ramond fields both theories spawned werequite different though, being one- and three-form gauge fields for the Type IIA theory,and zero-, two- and self-dual four-form gauge fields for the Type IIB theory.We omitted a lot <strong>of</strong> the technical subtleties <strong>of</strong> superstring theory. In particular, wedid not go deeply into supersymmetry, and completely disregarded the super-Virasoroalgebra, spawned by the modes <strong>of</strong> the energy-momentum tensor and supercurrent expansions.Readers dissatisfied and/or annoyed by such a lack <strong>of</strong> detail and rigour areurged to consult the given reference works cited in the introductory chapter to remedythis ailment. Complementing these, a good number <strong>of</strong> equally good review papers arewidely available on the internet, some <strong>of</strong> which are also listed in the bibliography.

Chapter 4Conformal Invariance“This has not, however, stopped [Disaster Area’s] earningsfrom pushing back the boundaries <strong>of</strong> pure hypermathematics,and their chief research accountant has recentlybeen appointed Pr<strong>of</strong>essor <strong>of</strong> Neomathematics atthe University <strong>of</strong> Maximegalon, in recognition <strong>of</strong> bothhis General and his Special Theories <strong>of</strong> Disaster AreaTax Returns, in which he proves that the whole fabric <strong>of</strong>the space-time continuum is not merely curved, it is infact totally bent.”Douglas AdamsHitchhiker’s Guide to the GalaxyInstead <strong>of</strong> focussing on the mathematical structure <strong>of</strong> a conformal field theory (CFT),this chapter aims to bring forward some <strong>of</strong> the main ideas behind it, and to motivateits usefulness, if not necessity, in the case <strong>of</strong> string theory.The key issue is that so far, we have only considered flat space-times; 26-dimensionalfor bosonic string theory and 10-dimensional for superstring theory. We are howeververy interested in knowing what happens when we consider curved space-times. Notonly because the dimensionality <strong>of</strong> previously mentioned space-times exceeds our own“visible” space-time, meaning ways need to be found to make those extra dimensions“invisible,” a popular way <strong>of</strong> doing so being to curl them up, but also because we knowfor a fact that our own beloved four-dimensional space-time is in fact curved itself. Itturns out that describing strings in general space-time backgrounds while requesting thatthe same symmetries we have in flat space-time are preserved, induces some remarkableconstraints on the allowed world sheet theories. These constraints however do not onlycome from the curved space-time, but also from the presence <strong>of</strong> background fields inbosonic string theory as well as both Type II theories (the ones we are interested in).To see where these background fields come from, recall that upon analysing thespectrum <strong>of</strong> the quantized bosonic and Type II closed strings, we found several masslessparticles they all had in common. Since these are quantum field theories, it is logical53

CHAPTER 4. CONFORMAL INVARIANCE 54to expect that these particles are the quantized products <strong>of</strong> classical fields. Hence, wemight raise the question if we should not have considered these fields, and their effectson the string action, before we quantized our string action, which is exactly what thischapter investigates.For all intended purposes <strong>of</strong> this chapter, it suffices to look at this problem for thespecific case <strong>of</strong> bosonic string theory.4.1 Conformal Field Theory Jr.Before anything else, it should again be duly noted that the aim <strong>of</strong> this section is not toexplain things, but merely to get a point accross, the point being that conformal fieldtheory is an absolute must for string theory.With that in mind, let us move on. Considering a general coordinate transformationx −→ x ′ , one finds that the metric changes according tog µν (x) −→ g ′ µν(x′ ) = ∂xα∂x ′µ ∂x β∂x ′ν g αβ (x) . (4.1)The group <strong>of</strong> conformal transformations is comprised <strong>of</strong> the subset <strong>of</strong> these generalcoordinate transformations that preserve the angle between two vectors. Concretely, thismeans that they leave the metric invariant, up to a rescaling. 1 Hence, when applying aconformal transformation, the metric changes according tog µν (x) −→ g µν′ (x′ ) = Ω(x)g µν (x). (4.2)The transformations that obey this property are translations, rotations, dilatations, andthe so-called special conformal transformations which vary a vector x µ according toδx µ = b µ x 2 − 2x µ b · x, (4.3)with b µ an infinitesimal translation. If one specializes to the case <strong>of</strong> Minkowski metric,and considers an infinitesimal coordinate transformation x µ −→ x µ + ǫ µ , one can findthe constraint on the parameter in flat space-time to be(δ µν + (D − 2)∂ µ ∂ ν ) ∂ · ǫ = 0, (4.4)with D the dimensionality <strong>of</strong> space-time. For D > 2, the parameter ǫ can be at mostquadratic in x, but if D = 2, as is the case for the string world sheet, this constraint doesnot apply. The consequence there<strong>of</strong> is that the associated generator algebra is infinitedimensional. To see this, and focussing on the string world sheet, introduce coordinatesz = τ − iσ, (4.5)¯z = τ + iσ. (4.6)1 Note that this is not the same as a Weyl rescaling, which does not entail a change <strong>of</strong> coordinates.

CHAPTER 4. CONFORMAL INVARIANCE 55One finds, after considering the right change in the metric, that∂ z¯ǫ = 0 = ∂¯z ǫ, (4.7)which is really great because this implies that ǫ = ǫ (z), and likewise ¯ǫ = ¯ǫ (¯z). Thismeans that we can expandǫ (z) = −+∞∑n=−∞a n z n+1 , (4.8)and a similar expression for ¯ǫ. This introduces an infinite amount <strong>of</strong> generators l n =−z n+1 ∂ z , in term giving rise to an infinite amount <strong>of</strong> conserved charges. Conservedcharges amount to restrictions on the theory, which is good news, because the moreconstrained a theory is, the more powerful it is. Added to this is the fact that byintroducing complex coordinates, or if you want by formulating the world sheet theoryon the complex plane, one can use all powerful tools <strong>of</strong> complex analysis, even more soif we redefine our coordinates z and ¯z asz = e τ−iσ , (4.9)¯z = e τ+iσ . (4.10)These coordinates map the entire string to concentric circles with radius e τ . We see thatthe infinite past on the world sheet corresponds to the origin, and as time increases, sodoes the radius <strong>of</strong> the circle. When one quantizes this theory, the equal time commutatorsthus become equal radius commutators, hence the name radial quantization.The name “conformal field theory” suggests that an object called conformal fieldshould exist. Indeed, it does, and specializing to the case at hand it is defined as a fieldφ(z, ¯z) that transforms under conformal transformations as 2( ) ∂zφ(z ¯z) −→ φ ′ ′ h ( ∂¯z′)¯h(z, ¯z) =φ ( z ′ (z) , ¯z ′ (¯z) ) . (4.11)∂z ∂¯zThe numbers h and ¯h are called the conformal weights <strong>of</strong> φ. Notice that conformalfields transform as tensors. As such, a conformal field theory is a field theory whichobeys conformal invariance, i.e. which is invariant under conformal transformations. Inparticular, it is a scale invariant theory, implying vanishing β-funtions.Talking about tensors, a very important one is the energy-moment tensor. As it turnsout, when one formulates the world sheet theory as a (quantum) conformal field theoryusing the coordinates as in Eqs. 4.9 and 4.10, under finite conformal transformationsthis tensor transforms as( ) ∂z′ 2(T zz (z) = T z∂z′ z ′ z′ ) + c12 D ( z ′) z , (4.12)2 Also quasi-conformal fields exist, which transform the same, but only under the so-called restrictedconformal group, which is the conformal group minus rotations.

CHAPTER 4. CONFORMAL INVARIANCE 56where D (z ′ ) zstands for the Schwarzian derivative, 3D ( z ′) z = ∂ z (z ′ )∂z 3 (z ′ ) − 3 (2 ∂2z z ′) 2(∂ z z ′ ) 2 . (4.13)The important thing is that we are now faced with a conformal anomaly c, <strong>of</strong>ten alsocalled central charge, a term that signals the breakdown <strong>of</strong> conformal invariance forgeneral conformal transformations in the quantum theory if c ≠ 0. The consequencehere<strong>of</strong> is that the trace <strong>of</strong> the energy-momentum tensor no longer vanishes, but nowbecomesT α α = − c12 R(2) , (4.14)with R (2) representing the Ricci scalar <strong>of</strong> the world sheet. This implies that we willrun into problems when considering non-flat world sheet space-times, or at least severerestrictions as to the possible consistent configurations.On a final note for interested and/or intruiged parties, all one could ever wish toknow about conformal field theory, and then some, can be found in [4].4.2 <strong>String</strong> Theory vs. QFT vs. General RelativityRefreshing the Polyakov action we used in order to study the bosonic string (Eq. 2.5),S σ = − 1 2 T ∫d 2 σ √ −hh αβ η µν ∂ α X µ ∂ β X ν ,we see that a possible generalization to curved backgrounds naturally imposes itself,namelyS g = − 1 ∫4πα ′ d 2 σ √ −hh αβ g µν (X)∂ α X µ ∂ β X ν , (4.15)where we have simply replaced the Minkwoskian metric with an arbitrary, non-trivial,field-dependent metric. One might argue that this is actually using results before havingobtained them, as the g µν field came about by constructing the spectrum <strong>of</strong> the quantizedclosed string (or if you prefer: is generated by the strings themselves), but it can beshown without great difficulty 4 by considering perturbations around η µν that this actiondoes indeed comply with the way the graviton arises in string theory.That seems easy enough, but we have seen in the previous chapters that the bosonicstring theory, as well as the (NS,NS) sector <strong>of</strong> both Type II superstring theories containnot one, but three massless bosonic fields: 5 the space-time metric g µν which we just3 Note that it seems like everyone uses their own little symbol to represent this function. We follow[14].4 You might ask yourself: “So why don’t you do it here?” The answer is that this short demonstrationdeals with vertex operators, and hence string interactions, which the author <strong>of</strong> this thesis has purposfullyleft out <strong>of</strong> his writings.5 The Type II theories also contain massless fields in the other sectors as we saw <strong>of</strong> course, but thefields originating from the (NS,NS) sector are the only ones they have in common with each other, andwith the bosonic theory.

CHAPTER 4. CONFORMAL INVARIANCE 57dealt with, the anti-symmetric Kalb-Ramond field B µν and the dilaton field Φ. These aresometimes referred to as the universal factor. So we should also find a way to incorporatethese last two fields in our action since they are equally part <strong>of</strong> the background. Ourprevious recipe can be applied without too much difficulty to the B µν field, 6S B = − 14πα ′ ∫d 2 σǫ αβ B µν (X) ∂ α X µ ∂ β X ν , (4.16)in which ǫ αβ is the usual anti-symmetric tensor density with components ǫ 01 = −ǫ 10 = 1and ǫ 00 = ǫ 11 = 0, but the dilaton coupling is considerably more subtle. The right wayto incorporate it turns out to beS Φ = − 14π∫d 2 σ √ −hΦ(X) R (2) , (4.17)in which R (2) again represents the world sheet Ricci scalar. Notice that this action isone order higher in α ′ than Eqs. 4.15 and 4.16. The full updated Polyakov action thusbecomesS σ = S g + S B + S Φ = − 14πα ′ ∫− 14π∫( √−hh )d 2 σαβ g µν + ǫ αβ B µν ∂ α X µ ∂ β X νd 2 σ √ −hΦ(X)R (2) . (4.18)Unfortunately, some bad news awaits us (yet again). This updated form still preservesparameterization invariance, and classicaly, both S g and S B conserve Weyl invariance,but S Φ does not. 7 This is bad news, as we needed Weyl invariance to be able to gaugefix our metric, in turn to be able to quantize our theory. Moreover, when consideringEq. 4.18 as describing a quantum field theory, Weyl invariance gets broken (for nontrivialmetrics) in general due to quantum effects. Recall that a consequence <strong>of</strong> Weylinvariance was the tracelessness <strong>of</strong> the enery-momentum tensor. Hence, breakdown <strong>of</strong>Weyl invariance will be reflected in the trace <strong>of</strong> T αβ . More generally speaking, breakdown<strong>of</strong> scale invariance in quantum field theories is usually associated with the non-vanishing<strong>of</strong> the β-function. Calculations <strong>of</strong> the less enjoyable variety based on an α ′ powerexpansion <strong>of</strong> Eq. 4.18 reveal thatT α α = − 12α ′βg µνh αβ ∂ α X µ ∂ β X ν − 12α ′βB µνǫ αβ ∂ α X µ ∂ β X ν − 1 2 βΦ R (2) , (4.19)6 Some authors include a complex factor i in this action.7 To be exact, it does not except when Φ is constant. In that case, S Φ ∼ χ M = − 14πRd 2 σ √ −hR (2) ,which is the Euler characteristic <strong>of</strong> the manifold. This is a topological invariant, and thus unaffected byWeyl rescalings.

CHAPTER 4. CONFORMAL INVARIANCE 58in which 8β g µν = α ′ (R µν + 2∇ µ ∇ ν Φ − 1 4 H µκσH κσν)+ O ( α ′2) , (4.20)(βµν B = α ′ − 1 )2 ∇κ H κµν + ∇ κ ΦH κµν + O ( α ′2) , (4.21)( D − 26β Φ = α ′ 6α ′ − 1 2 ∇2 Φ + ∇ κ Φ∇ κ Φ − 1 )24 H κµνH κµν + O ( α ′2) , (4.22)where we usedH κµν = ∂ κ B µν + ∂ µ B νκ + ∂ ν B κµ .These are the generalized Einstein equations for the fields under consideration. Notethat the expansion in α ′ is only valid if the typical length scale <strong>of</strong> space-time (usuallyexpressed in terms <strong>of</strong> the Ricci scalar) is much larger than α ′ , or equivalently if spacetimeis only slightly curved at the string length scale (recall that l 2 s = 2α ′ ). The actionfor the scalar field, being one order higher in α ′ , can be considered as a first orderperturbative correction.Demanding that Weyl invariance is still respected implies that all above mentionedβ-functions should vanish, in turn demanding that the world sheet theory is describedby a conformal field theory, i.e. a scale invariant theory. It should be noted that whenβ g = β B = 0, we are left withT α α = − 1 2 βΦ R (2) , (4.23)which as we saw is exactly the conformal anomaly that arises when one quantizes atwo-dimensional conformal field theory. Hence, when β g = β B = 0, Eq. 4.18 describesa conformal field theory with central charge equaling β Φ . 9 Since β Φ should also vanish,we are left with an anomaly free conformal field theory.Furthermore, it is worth pointing out that when all above β-functions are set to zero,what remains are second order differential equations for the background fields. As itturns out, these can be regrouped into one single action,S = 12κ 2 0∫d D X √ −Ge −2φ [R + 4∇ µ Φ∇ µ Φ − 1 12 H µνλH µνλ2 (D − 26)−3α ′ + O ( α ′)] , (4.24)in which κ 0 represents a constant with no physical significance, as it can be absorbed ina redefinition <strong>of</strong> Φ. Varying this action with respect to the background fields results insaid differential equations. Remark that from this, if one splits Φ into its expectationvalue Φ 0 and the deviation there<strong>of</strong>, Φ −→ Φ 0 + Φ, one can identify g s = e Φ 0. In otherwords, the expectation value <strong>of</strong> the scalar field Φ defines the string coupling constant.If so wanted, one could redefine the metric G µν −→ ˜G µν by multiplying with a suitable8 Readers curious as to the details <strong>of</strong> the derivations <strong>of</strong> these β-functions are referred to §2.1.3 in [5].9 It can be shown that when β g = β B = 0, β Φ is constant.

CHAPTER 4. CONFORMAL INVARIANCE 59power <strong>of</strong> e Φ in order to bring the gravitational part <strong>of</strong> the action to the standard Hilbertform 1 ∫2κ d X√D − ˜GR, allowing one to identifyκ ≡ κ 0 e Φ 0= √ 8πG N , (4.25)with G N Newton’s gravitational constant. Details can be found in §3.7 in [18] or §2.7in [12].4.3 SummaryThe few preceding pages attempted to highlight some <strong>of</strong> the essential characteristics <strong>of</strong>a conformal field theory. Then a short review was given <strong>of</strong> the problems associated withthe generalization <strong>of</strong> (bosonic) string theory from flat space-time to curved space-times,further including the Kalb-Ramond and scalar Φ background fields, and it was shownthat the resolution <strong>of</strong> these problems demanded that the world sheet theory be describedby a conformal field theory. It was further brought forward that the expectation value<strong>of</strong> the Φ field in fact determines the string coupling g s .

Chapter 5Branes“Many speak <strong>of</strong> the legendary and gigantic starship Titanic,a majestic and luxurious cruise liner [which] hadthe misfortune to be built in the very earliest days <strong>of</strong> ImprobabilityPhysics, long before this difficult and cussedbranch <strong>of</strong> knowledge was fully, or at all, understood. Thedesigners and engineers decided, in their innocence, tobuild a prototype Improbability Field into it, which wasmeant, supposedly, to ensure that it was Infinitely Improbablethat anything would ever go wrong with anypart <strong>of</strong> the ship. They did not realize that because <strong>of</strong> thequasi-reciprocal and circular nature <strong>of</strong> all Improbabilitycalculations, anything that was Infinitely Improbable wasactually very likely to happen almost immediatly.”Douglas AdamsHitchhiker’s Guide to the GalaxyDespite its name, string theory does not exclusively handle about strings. Other physicalobjects also inhabit this theory, amongst which we find the D-branes and NS5-branes,both being special cases <strong>of</strong> general p-branes: objects that extend in p dimensions <strong>of</strong>space. This chapter aims to introduce these two protagonists, concentrating on theType II superstring theory case, since that is the one that concerns us in the carried outresearch.5.1 Dp-branesA Dp-brane is a physical object that lives in D-dimensional space-time, and extendsin p spatial dimensions (with p < D), and hence describes a (p + 1)-dimensional worldvolume. One does not always specify the p, in which case one simply talks about D-branes. The defining property <strong>of</strong> a D-brane is that open strings with Dirichlet boundary60

CHAPTER 5. BRANES 61conditions can end on them; i.e. that the endpoints <strong>of</strong> an open string satifsying Dirichletboundary conditions are located on a D-brane. 1What makes D-branes so extremely interesting is that they provide a way to introducegauge symmetries in string theory, which in turn hints at a possible way to createstandard model-like models from a stringy perspective.The easiest way to motivate the necessity for D-branes is by considering T-duality.Hence, this section will start with a short exposition <strong>of</strong> this subject. Then we will have alook at the influence <strong>of</strong> the gauge fields generated by the closed strings in the two TypeII superstring theories on the possible D-branes in these theories. We will conclude byhighlighting a possible low-energy action that describes the world volume <strong>of</strong> a D-brane.5.1.1 T-dualityTo make our lives easier, we will again go back for a second to the bosonic string theoryand its corresponding 26-dimensional space-time. We will investigate what happenswhen we compactify one spatial dimension. Compactifying in essence comes down totaking a dimension, and identifying (at least) two points in this dimension with eachother. A natural way <strong>of</strong> doing so, is to consider this dimension to be curled up into acircle.We will start by considering what happens to closed strings, and then have a lookat what changes for the case <strong>of</strong> open strings.Closed stringsThe first thing to do, is simply to choose one <strong>of</strong> the 25 spatial dimensions to compactify.We will not try to be original, but instead choose to do as everybody else, and hencechoose the 25 th dimension. As such, compactifying this dimension amounts to statingthat this dimension satisfies the topology <strong>of</strong> a circle with some arbitrary (but fixed, <strong>of</strong>course) radius R. Now, a closed string with a component that lives in this dimensionobviously has to go around the entire circle at least once. But nothing forbids it <strong>of</strong>winding around it multiple times. Just think <strong>of</strong> an elastic that you wind around acurled up poster or something: you can wrap it around just once, but nothing forbidsyou from making a figure-<strong>of</strong>-eight with your elastic and hence wrapping it around twice.Because <strong>of</strong> this newly acquired freedom, our original closed string boundary condition,Eq. 2.16, gets modified with an extra term as follows:X 25 (σ + π, τ) = X 25 (σ, τ) + 2πRW. (5.1)Herein, W ∈is called the winding number, and expresses how many times the stringwinds around the circular dimension. The direction in which the string is winded isencoded in the sign <strong>of</strong> W. This extra term is explained by the simple observation 2that the closed string boundary condition relates one and the same point on our string,1 This immediatly implies that a string and a D1-brane are two separate things.2 The simpler an observation, the longer it usually takes to figure it out.

CHAPTER 5. BRANES 62but if the string is wound a number <strong>of</strong> times then simply stating that X 25 (σ + π, τ) =X 25 (σ, τ) would identify the wrong points!The next thing to do is <strong>of</strong> course to take a look at the mode expansion <strong>of</strong> our string,and see if anything has changed. The first thing to note is that nothing changes in the24 spatial dimensions that were left uncompactified, and hence the mode expansion inthese dimensions also does not change. In the 25 th dimension however, for exactly thesame reason mentioned just earlier, we should add an extra term to account for thewinding <strong>of</strong> the string. The second thing to note is that compactifying does not changeanything to the oscillator modes. And the third and final thing to note is that due tothe compactification, the momentum in the 25 th dimension becomes quantized. In orderto see this, recall e.g. the case <strong>of</strong> Kaluza-Klein compactification <strong>of</strong> a free massless scalarfield ψ (x µ , y) living in a D +1-dimensional Minkowski space <strong>of</strong> which one dimension, y,is compactified, satisfying the massless Klein-Gordon equation( + ∂2y)ψ = 0, (5.2)where = ∂ µ ∂ µ with µ ∈ {0, . . .,D − 1}. Given that y is periodic, so is the Fourier expansionin this dimension, and hence as a general solution to the Klein-Gordon equationwe findψ (x µ , y) =+∞∑n=−∞e i n R y ψ k (x µ ), (5.3)in which R is the radius <strong>of</strong> the compactified dimension, and e i n R y = e ipyy , implying thatp y = n R . (5.4)In the case <strong>of</strong> the string, the same argument holds (recall that the X µ ’s are scalar fields),but with this distinction that we definep 25 = K R , (5.5)where we have defined the Kaluza-Klein excitation number K ∈. Taking all this intoaccount, and starting from the generic solutionX 25 (σ, τ) = x 25 + 2α ′ p 25 τ + 2RWσ + . . ., (5.6)splitting in left- and right-movers leaves us withXR 25 (τ − σ) = 1 (x 25 − ˜x 25) +(α ′K )2R − WR (τ − σ) + oscillators,XL 25 (τ + σ) = 1 (x 25 + ˜x 25) +2(α ′K R + WR )(τ + σ) + oscillators, (5.7)where ˜x 25 is some arbitrary constant that cancels in the sum, which will prove itselfvery useful in a second.

CHAPTER 5. BRANES 63We should again define α 0 and ˜α 0 operators in such a way that we can “completethe sum” when we take the derivatives <strong>of</strong> the expansions. To this end, we define√2α ′ α0 25 = α ′K R− WR, (5.8)√2α′˜α 250 = α ′K R+ WR. (5.9)Now that we have these at our disposal, recall the (still unchanged) condition expressedin Eq. 2.99, as well as the defintion <strong>of</strong> L m in Eq. 2.97, and defineM 2 = −24∑µ=0p µ p µ , (5.10)i.e. the mass in the 25-dimensional non-compact space-time. Using all this, we find that0 = L 0 − 1,⎡= 1 ⎣ ( α 25 ) 224∑0 + α µ 02α 0µ +µ=0= ˜L 0 − 1,⎡= 1 )⎣ 25 224∑ (˜α 0 + ˜α µ 02˜α 0µ +µ=025∑+∞∑ν=0 n=125∑+∞∑ν=0 n=1α ν −nα nν⎤⎦ − 1,˜α ν −n˜α nν⎤⎦ − 1. (5.11)Remembering that for the closed bosonic string we defined α µ 0 = 1 2 l sp µ (but now withµ < 25) and definingthe above tells us thatN R =N L =25∑∑α−nα ν nν , (5.12)ν=0 n>025∑∑˜α −n˜α ν nν , (5.13)ν=0 n>012 α′ M 2 = ( α025 ) 2+ 2NR − 2 =(5.14))25 2 (˜α 0 + 2NL − 2. (5.15)Filling in Eqs. 5.8 and 5.9, and taking sum and difference allows us to conclude that[ (αN R − N L = −1 ′ )K 2 ( α ′ ) ]4α ′ R − WR K 2−R + WR= KW, (5.16)

CHAPTER 5. BRANES 64meaning that the level-matching condition got somewhat altered, and[ (K ) 2 ( ) ]WR2α ′ M 2 = α ′ +R α ′ + 2N L + 2N R − 4. (5.17)This last equation hides a symmetry, namely it is invariant under the simultaneoustransformations {R −→ ˜R = α ′ R −1 ,(5.18)W ←→ K.This symmetry is exactly what is called T-duality, and it teaches us that for closedstrings compactification on a circle <strong>of</strong> radius R is physically equivalent to, and evenindistinguishable from, compactification on a circle <strong>of</strong> radius α ′ R −1 . You might wonderwhy it is called a duality and not a symmetry. The reason is simple: it is only asymmetry in the closed string case, but not in the open string case as we will soon findout. However, we will still be able to map solutions <strong>of</strong> one theory to solutions <strong>of</strong> another,hence we are left with a duality. According to [11] (p.38), the “T” stands for “targetspace,” being as we map different space-time configurations into one another.At any rate, the effect <strong>of</strong> performing a T-duality transformation on the closed stringalso entails that one changesX 25R −→ −X 25R ; X 25L −→ X 25L . (5.19)This mapping is obvious for the zero modes, in which case it simply follows from Eqs.5.8 and 5.9, but for the other modes we need to impose this. We can do this, as thischanges nothing to the physics <strong>of</strong> our theory. This map leaves all physical quantities<strong>of</strong> the theory unchainged, and hence truely constitutes a symmetry. This allows one toexpress the theory using an equivalent coordinate ˜X defined aswith the generic expansion˜X 25 (σ, τ) = X 25L (τ + σ) − X 25R (τ − σ), (5.20)˜X 25 (σ, τ) = ˜x 25 + 2α ′K R σ + 2RWτ25 + oscillators. (5.21)We note that R can take on a specific value at which the theory is self-dual, namelyR = √ α ′ . At this radius, the massless closed string spectrum gets modified so as toacquire two SU(2) gauge groups, one left-moving and one right-moving. For a shortdiscussion, see [12] §3.3.It is worth pointing out a major difference between Kaluza-Klein (KK) compactificationand T-duality. With KK compactification, what one does is compactify adimension, so that one can make it “disappear” by choosing it infinitally small (R −→ 0limit), in which case its physics decouple from that in the other dimensions. This ispossible, because in KK theory, one only has a mass-squared-like factor going as ∼ R −2 .With T-duality however, two mass-squared-like terms appear, one going as ∼ R −2 andanother going as ∼ R 2 . You simply cannot make them both disappear simultaneously!

CHAPTER 5. BRANES 65<strong>Open</strong> stringsNext, we turn ourselves to the case <strong>of</strong> the open string. We consider an open string obeyingNeumann boundary conditions in all 25 spatial dimensions, for which the expansionwas given by Eq. 2.41, which we refresh:X µ (σ, τ) = x µ + l 2 sp µ τ + il s∑n≠01n αµ ne −inτ cos (nσ).Again compactifying the X 25 dimension, this time we do not need to include any specialextra terms as in the closed string case. It remains true however that the p 25 momentumcomponent becomes quantized, so that p 25 = nR −1 with n ∈. Further recalling thatwe defined α µ 0 = l sp µ , we find for the expansion <strong>of</strong> X 25 :∑X 25 (σ, τ) = x 25 + l s α0 25 1τ + il sn α25 n e −inτ cos (nσ). (5.22)Although we did not do so before because it did not bare any interesting features withit at the time, we will now also expand this into left- and right-movers. We may do so,as we can consider the oscillations <strong>of</strong> an open string to be superpositions <strong>of</strong> left- andright-movers that combine into standing waves. We will do so, because <strong>of</strong> course, wewould like to be able to formulate the theory using a “dual coordinate” ˜X 25 , just as inthe closed string case. This expansion amounts ton≠0XR 25 (τ − σ) = x25 − ˜x 25+ 1 2 2 l sα0 25 (τ − σ) + i l ∑s 12 n α25 n e −in(τ−σ) , (5.23)n≠0∑XL 25 (τ + σ) = x25 + ˜x 25+ 1 2 2 l sα0 25 (τ + σ) + i l s2n≠01n α25 n e −in(τ+σ) , (5.24)where again ˜x 25 is an as-<strong>of</strong>-yet arbitrary constant that cancels in the sum. The biggestdifference with the closed string case, is that this time there is only one set <strong>of</strong> modes (i.e.,there are no ˜α’s). T-dualizing by sending R −→ ˜R = α ′ R −1 will again send X R −→ −X Rand X L −→ X L , and so we define the dual coordinate ˜X 25 (σ, τ) = X L − X R , whichexplicitely becomes∑˜X 25 (σ, τ) = ˜x 25 + l s α0 25 1σ + l sn α25 n e −inτ sin(nσ). (5.25)We see that the zero-mode gets multiplied by σ instead <strong>of</strong> τ, as was the case for X 25 ,and that the cos (nσ) got exchanged for −i sin(nσ). This hints at something that turnsout to be true, namelyn≠0∂ σ X 25 = ∂ τ ˜X25 , (5.26)∂ τ X 25 = ∂ σ ˜X25 , (5.27)

CHAPTER 5. BRANES 66i.e. our original Neumann boundary conditions got transformed into Dirichlet boundaryconditions! Had we started out with Dirichlet boundary conditions, it is easy to seefrom Eq. 2.39 that the same would have happened but the other way around, i.e. wewould have found that D boundary conditions got turned into N boundary conditions.Furthermore, we note that˜X 25 (π, τ) − ˜X 25 (0, τ) = 2π ˜Rn, (5.28)meaning our string has to wrap an integer time around the T-dualized circle. This isquite an interesting observation, given that if a string has Neumann boundary conditionsin some (compact) direction, it can not have a winding number in that direction.Intuitively this is clear: the string can move freely in that direction, so nothing can forceit to keep its endpoints fixed. But since it can move, it also gains momentum. In theT-dual picture, this observation gets inversed: the string endpoints become fixed, so itcan wind (and as Eq. 5.28 shows, it is even forced to), but since it cannot move, itcannot gain any momentum.The crucial point in all this, is that as far as closed strings are concerned, thereis no physically significant difference between compactification on a circle <strong>of</strong> radius R,or a circle <strong>of</strong> radius ˜R = α ′ /R. For open strings however, this is not the same, andthat is something we do not like, because T-duality seemed like a really nice idea andvery powerful tool. So how can we fix this awkward little problem? Answer: introduceD-branes! So in order to make sense <strong>of</strong> this picture, it has been proposed (and thisproposition has since been developped in full glory) that an open string with Dirichletboundary conditions does not have its endpoints fixed to some arbitrary points in spacetimethat somehow are more special than their neighbours, but instead that its endpointslie on the surface <strong>of</strong> a physical object that lives in (p + 1)-dimensional space-time. Theendpoints <strong>of</strong> the string are free to move on this surface, but are bound to remain fixed indirections perpendicular to it. This object is called a Dp-brane, with the D referencingto Dirichlet for obvious reasons. So in the case <strong>of</strong> our original string that had Neumannboundary conditions in all 25 spatial dimensions, we can now interpret this as an openstring whose endpoints live on the surface <strong>of</strong> a space-time-filling D25-brane. Since thesurface <strong>of</strong> this D-brane comprises all <strong>of</strong> space-time, the open string is free to move as itpleases. When we compactify the X 25 direction and T-dualize it however, what happens“underneath the surface” is that we remove one dimension <strong>of</strong> our D-brane, making it aD24-brane that does no longer extend in the X 25 direction. Hence, since this brane stilllives in a 25 spatial-dimensional world, its position must be fixed in the X 25 dimension, 3which explains why suddenly our open string gains Dirichlet boundary conditions in thisdirection and is forced to wrap an integer number <strong>of</strong> times around it.This idea can easily be generalized to compactification <strong>of</strong> more than one spatialdimension. Remark however that because an open string wraps around a circular direction,and hence has both its endpoints identified in this direction, does not make ita closed string! The endpoints can still take on different values in the other directions.3 Consider e.g. a disc in the x − y plane that lives in a 3D world. If this disc cannot move in the zdirection, when we project it onto the z-axis it is bound to take on a fixed value.

CHAPTER 5. BRANES 67Also remark that it are only the endpoints <strong>of</strong> the open string that are fixed; all otherpoints on the string are still free to move as they please.D-branesAs we have just introduced them, Dp-branes are physical objects that extend in p spatialdimensions, and whose defining property is that the endpoints <strong>of</strong> open strings satisfyingDirichlet boundary conditions can (or even have to) end on them. We see that in essence,D-branes were present right from the start, only betraying their presence through theDirichlet boundary conditions which first got dismissed as being unphysical due to thefact that they break Poincaré invariance. As a consequence, they only grew in popularitywhen T-duality was discovered in 1989.As an intuitive argument 4 as to why these objects should behave like physical objects,consider this next scenario. An open string fixed to a D-brane moves around untilsomehow both its endpoints meet. As <strong>of</strong> that moment, the string becomes a closedstring, and hence is no longer bound to the surface <strong>of</strong> this D-brane. Instead, it can get<strong>of</strong>, and roam around in space-time, enjoying the sight. But at some point, it might hitanother D-brane, at which time it can “open up” again, and become fixed to the surface<strong>of</strong> this other D-brane. In this case, two D-branes will have exchanged a (closed) string,and hence, talked to each other. Only physical objects can interact, hence implying thatD-branes should indeed be physical.Something crucial happens to the spectrum <strong>of</strong> open strings with Dirichlet boundaryconditions. We now know that if a string has Neumann conditions in some directions, letus say 0 to p (with 1 ≤ p < 25), and Dirichlet in the others, this amounts to saying thatthis string has endpoints lying on a p-brane. We can now denote the Neumann directionswith an index n and the Dirichlet directions with an index d, i.e. X µ = { X n , X d} withn ∈ {0, . . .,p} and d ∈ {p + 1, . . .,25}. The next step is to combine X 0 and X 1 intolight-cone coordinates, so that we can use the light-cone gauge to construct the spectrum<strong>of</strong> this string. After we do this, we are left with (p − 1) transverse Neumann directions,and thus (p − 1) α−1 i operators (i ∈ {2, . . .,p}) we can let act on our groundstate. Ifwe use these operators, we create a massless state, just as we saw in Chapter 2, butthis time in (p − 1) <strong>of</strong> the dimensions in which our Dp-brane is stretched. We create aphoton that lives on the D-brane! Hence, every Dp-brane has a Maxwell field living onits world volume. The α−1 d operators generate states that from the point <strong>of</strong> view <strong>of</strong> theD-brane transform as scalars. All these fields are perpendicular to the D-brane worldvolume. They can be regarded as excitations <strong>of</strong> the D-brane itself.This last point gives rise to a nice way <strong>of</strong> eliminating tachyons from the open stringbosonic spectrum. It has been suggested that a space-time filling D25-brane is unstablebecause <strong>of</strong> the tachyon that lives on its world volume. Because <strong>of</strong> this, the D25-branewould actually decay, and the products <strong>of</strong> this decay would be closed strings. This isalso true for Dp-branes with p < 25. As a consequence, bosonic string theory couldvery well not contain any stable D-branes at all. As <strong>of</strong> yet, no way has been found toeliminate the closed string tachyon, though.4 This argument has been taken from [5].

CHAPTER 5. BRANES 68The presence <strong>of</strong> a Maxwell field on the D-brane has a very important consequence.Consider the countour integral ∮⃗E · d ⃗ l. (5.29)CWhen we consider the electric field to be static, and this contour to be lying somewhere inordinary space, we can invoke Stokes’ theorem in combination with the (static) Maxwellequation ∇ × ⃗ E = 0 to obtain∮C∫⃗E · d ⃗ l =S(∇ × E ⃗ )d⃗a = 0. (5.30)When considering compact dimensions, something fundamental changes, namely thecontour no longer bounds a surface since there is no “inner space” anymore. As aconsequence, Eq. 5.29 can take on non-zero values. Generalizing to the case <strong>of</strong> a vectorpotential A µ , and considering its component along the compact dimension (say x 25 ),defineW q = e iq H c dx25 A 25. (5.31)This quantity is called a Wilson line, and it takes on a constant value in the compactinterval [0, 2π]. Now we can consider a setup with several parallel Dp-branes, say N<strong>of</strong> them. We can thus consider several different open string configurations, in generaldefined by two labels i and j, respectively labeling the Dp-brane on which the stringstarts (σ = 0) and ends (σ = π). Now, suppose one direction is compactified, andthat our Dp-branes wrap around this direction. Due to the Maxwell field living on theD-branes, they will acquire a Wilson line θ i along the compact dimension. Amazinglyenough, when we take the T-dual <strong>of</strong> this compact dimension, the resulting D(p − 1)-branes will position themselves on the dual circle at angles θ i . This means that anopen string lying on branes i and j in the original description will stretch on the dualcircle between angles θ i and θ j . The labels i, j are called “Chan-Paton charges.” Thismechanism introduces gauge groups as follows: k coincident branes give rise to a U(k)group, while l seperate branes make for a U(1) l group.5.1.2 D-branes in Type IIA and Type IIB theoriesRecall the coupling <strong>of</strong> the string to the Kalb-Ramond field shown in Eq. 4.16,S B = − 1 ∫4πα ′ d 2 σǫ αβ B µν (X) ∂ α X µ ∂ β X ν .This is in fact a generalization <strong>of</strong> the coupling <strong>of</strong> a vector field to a gauge field,∫dx µS A = q dτA µdτ , (5.32)where q represents the coupling constant to this field (e.g. electric charge). This generalizationimplies that a string couples electrically to the Kalb-Ramond field.

CHAPTER 5. BRANES 69Generally speaking though, considering a p-form gauge field A p , we can constructfrom it a field strength F p+1 , and a dual field strength by taking the Hodge dual ⋆F p+1which is a (D − p − 1)-form. Recalling the form formulation <strong>of</strong> Maxwell’s equations forthe Maxwell field A µ and associated field strength F = dA in the presence <strong>of</strong> sources,dF = ⋆J m ; d ⋆ F = J e , (5.33)in which J m represents the magnetic source and J e represents the electric source, wecan generalize this to a p-form gauge field. The important thing to note is that if abrane couples to such a field, it acquires a charge, and these charges can be computedby integrating the field strengths over a sphere surrounding the brane. So looking atthe electrical coupling, we see that in order to compute the charge, we would need tointegrate over a (D − p − 1)-sphere, and in the case <strong>of</strong> magnetic coupling over a (p + 1)sphere. Geometry tells us that a (D − p − 1)-sphere can surround a (p − 1)-dimensionalobject, while a (p+1) sphere surrounds a (D−p−3)-dimensional object. The importantconclusion to draw from all this, is that the dimensionality <strong>of</strong> the gauge field defines towhat type <strong>of</strong> brane it can couple.Having established all this, this immediatly begs the question if also D-branes coupleto any fields, and hence acquire charge under those fields? Considering the above,we could reformulate the question as: what gauge fields do we have at our disposal?The answer, as we saw in Chapter 3, depends on what type <strong>of</strong> superstring theory oneconsiders.Type IIAIn the Type IIA theory gauge fields A µ and A µνρ are present. Electrically, this couplesto Dp-branes with p ∈ (0, 2), and magnetically to Dp-branes with p ∈ (4, 6). It is worthpointing out that a D8-brane can actually also occur. This requires a nine-form gaugefield, which in ten-dimensional space-time is nondynamical. In short: the Type IIAtheory allows for stable Dp-branes with p even.Type IIBIn the Type IIB theory gauge fields A, A µν and A µνρσ are present. This couples electricallyto Dp-branes with p ∈ {−1, 1, 3}, and magnetically to Dp-branes with p ∈ {3, 5, 7}.The D(−1)-brane the zero-form gauge field couples to is called a D-instanton [13], anobject which is localized in space and time. Furthermore we see that a D3-brane appearstwice. This is actually the same brane twice; it is self-dual. In some cases, space-time fillingD9-branes can also occur. In short: the Type IIB theory allows for stable Dp-braneswith p odd.BPSAnother important feature <strong>of</strong> all D-branes present in both Type II theories is thatthey preserve half <strong>of</strong> the supersymmetries present in the theories without D-branes. A

CHAPTER 5. BRANES 70heuristic argument to make this claim somewhat more acceptable is that open stringscan in fact be regarded as being excitations <strong>of</strong> a D-brane world volume. <strong>Open</strong> stringshave only half the supersymmetries closed strings have, implying that also D-branespreserve only half <strong>of</strong> them. An important consequence <strong>of</strong> this is that D-branes exert n<strong>of</strong>orce on each other. They can exchange (closed) strings, and they attract each othergravitationally, but the net result <strong>of</strong> these interactions is zero. Often, one says that anobject satisfying this condition is a (half-)BPS state, where BPS stand for the namesBogomolny, Prasad and Sommerfield.5.1.3 T-duality in Type IIA and Type IIB theoriesWithout going into details, if we consider for example the compactification <strong>of</strong> the X 9direction in the Type II superstring theories, followed by carrying out T-duality in thisdirection, we will still obtain thatBy virtue <strong>of</strong> world sheet supersymmetry, alsoX 9 R −→ −X 9 R ; X 9 L −→ X 9 L. (5.34)ψ 9 R −→ −ψ 9 R ; ψ 9 L −→ ψ 9 L. (5.35)In other words, we reverse the chirality <strong>of</strong> the Ramond sector right-movers, because wealter the sign <strong>of</strong> the d 9 0 operator. As was pointed out earlier, this operator is related tothe Dirac Γ-matrices by the relation Γ µ = √ 2d µ 0 . Given that the chirality operator Γ11is defined asΓ 11 = Γ 0 Γ 1 . . .Γ 9 , (5.36)reversing the sign <strong>of</strong> one <strong>of</strong> these matrices effectively inverses the chirality. As such,T-duality maps the Type IIA theory into the Type IIB theory, and vice versa. If weT-dualize another dimension, we will again reverse chiralities, and come back to thetheory we started out with, and so forth.5.1.4 Dirac-Born-Infeld ActionTo describe the dynamics <strong>of</strong> our D-branes, we still need to concoct an action for them.A popular low-energy action is the so-called Dirac-Born-Infeld (<strong>DBI</strong>) action. We willnot derive this action in full detail, but merely sketch the derivation. Details can befound in e.g. [12] §4.2, and [13] §8.5.1.Before we devise this action, we can already start by guessing what different contributionswe need by comparing to the case <strong>of</strong> a string. So the first thing we would need,is a term that describes the embedding <strong>of</strong> the brane in the external space-time. Second,we will need to consider contributions from coupling to the B µν field (since D-braneslive in the same space-time as closed strings, and hence also feel the fields generated bythese last ones), as well as the action for the Maxwell field A µ generated by the openstrings ending on, and itself living on the D-brane.

CHAPTER 5. BRANES 71The embedding <strong>of</strong> the Dp-brane in space-time can be achieved, just as for the string,by first defining a set <strong>of</strong> coordinates σ n with n ∈ {0, . . .,p} on the brane. This allowsus to write the pullback <strong>of</strong> the space-time metric g µν to the brane, and write down thefirst part <strong>of</strong> our action asS p = −T p∫d p+1 σe −Φ √− det g µν (X)∂ α X µ ∂ β X ν , (5.37)where as usual ∂ α,β denotes derivation with respect to σ α,β . We need to include a factore −Φ ∼ gs−1 in this action because this is in fact an open string tree level action.Moving on to the B µν an A µ fields, recall that the Kalb-Ramond coupling <strong>of</strong> a stringwas given by Eq. 4.16 which we repeat for convenience,S B = − 1 ∫2πls2 d 2 σǫ αβ B µν (X)∂ α X µ ∂ β X ν , (5.38)and use that the coupling <strong>of</strong> A µ to the world sheet boundary ∂M is given by∫S A = dτA µ ∂ t X µ . (5.39)∂MIf we want to mimick this behaviour for the D-brane, we are not free to take anycombination as we please. Space-time gauge invariance imposes a choice on us. Thepoint is that S A is invariant under a space-time gauge transformation A µ −→ A µ +∂ µ f,but when applying a space-time gauge transformation B µν −→ B µν + ∂ µ Λ ν − ∂ ν Λ µ wepick up a boundary termδS B = − 1 ∫2πls2 dτΛ µ ∂ τ x µ . (5.40)∂MTo cancel this boundary term and restore gauge invariane, we see that A µ should transformsimultaneously according toA µ −→ A µ − 12πls2 Λ µ . (5.41)From this we can conclude that the only way to inlucde both fields in our action is byusing the combinationB µν + 2πα ′ F µν (5.42)which remains invariant under both symmetries.By considering the case <strong>of</strong> a D2-brane in Minkowski space-time, extended in directionsX 1 and X 2 (i.e. ( σ 0 , σ 1 , σ 2) = ( t, X 1 , X 2) ) <strong>of</strong> which one dimension is compactified,say X 2 , one can show that upon T-dualizing the compactified dimension, the remainingD1-brane is tilted with respect to the X 1 direction, namely the dualized direction X ′2satisfiesX ′2 = 2πα ′ X 1 F 12 , (5.43)

CHAPTER 5. BRANES 72where F 12 is the (12) component <strong>of</strong> the Maxwell field strength living on the D2-branewhich is taken to be constant. The tilting angle is given by θ = tan −1 (2πα ′ F 12 ). Because<strong>of</strong> this, the new pullback from the metric to the brane becomes⎡⎤−1 ∂ 1 X ′2 0 . . . 0∂ 1 X ′2 1 0 . . . 0g αβ =0 0 0 . . . 0(5.44).⎢ . . . .. . ⎥⎣⎦0 0 0 . . . 0resulting in the action for the D1-brane (we ignore the e −Φ factor)∫ √∫ √S D1 ∼ d 2 σ 1 + (∂ 1 X ′2 ) 2 = d 2 σ 1 + (2πα ′ F 12 ) 2 . (5.45)By boosting and rotating the D-brane, one can bring F µν in block-diagonal form, whichallows one to write the Dp-brane generalization <strong>of</strong> the above action as∫ √S Dp ∼ d p+1 σ − det(η µν + 2πα ′ F µν ). (5.46)This action is called the Born-Infeld action. By further applying T-dualizing arguments,one can obtain the full Dirac-Born-Infeld action which also incorporates non-trivialbackgrounds and the Kalb-Ramond field,∫S Dp = −T p d p+1 σe√− −Φ det(g αβ + B αβ + 2πα ′ F αβ ), (5.47)where indices αβ denote the pullbacks <strong>of</strong> the fields to the brane.The explicit form <strong>of</strong> T p can be computed by analysing the exchange <strong>of</strong> a closed loopbetween two parallel Dp-branes, and results inT p =Applying T-duality further results in a recursion relation,1(2π) p l p+1s g s. (5.48)T p−1 = (2πl s )T p . (5.49)On a final note, we point the reader to an alternative derivation <strong>of</strong> the <strong>DBI</strong> action,which consists <strong>of</strong> computing the β-function for the A µ gauge field, and solving itsequation <strong>of</strong> motion. Details can be found in [5], §2.3.3.1.

CHAPTER 5. BRANES 735.2 NS5-branesYou might be tempted into thinking that “NS-branes are branes on which strings withNeveu-Schwarz boundary conditions end.” This is however as far as possible from thetruth as could be. Matter <strong>of</strong> the fact is that strings cannot end on NS5-branes!We saw that a p-form gauge field can couple electrically to a (p − 1)-brane, andmagnetically to a (D − p − 4)-brane. We also saw that the string, which is a 1-brane,if a Kalb-Ramond field is present, couples electrically to this field. Hence, we shouldexpect that there is also a (D − 1 − 4) = 5-brane that couples to this field. This branedoes indeed exist (by which we mean that it is stable), and is exactly what is called theNS5-brane. It is called such because it arises from a gauge field which originates fromthe NS-NS sector in superstring theories. It is also sometimes referred to as the solitonicfivebrane. Remark that the B µν field is absent from Type I string theory due to the factthat this is an unoriented theory, and hence this theory contains no NS5-branes.NS5-branes are characterized by a string tension T NS5 that goes asT NS5 ∼ 1 gs2 . (5.50)Comparing this to the tension <strong>of</strong> the string (T s ∼ g 0 s) and D-branes (T Dp ∼ g −1s ) revealsthat the NS5-brane is much heavier than other types <strong>of</strong> branes at weak coupling. Anotherimportant property is that NS5-branes are also BPS states. As a consequence <strong>of</strong> this,they do not feel each other, or other BPS states such as D-branes, if they are combinedin such a way that the resulting configuration remains BPS.What makes them very interesting though is that although strings cannot end onthem, D-branes actually can. Which configurations are actually possible, and which arenot, is determined by principles <strong>of</strong> charge conservation. We will not go into this, butinterested parties are referred to [20] and [21]. The same principles <strong>of</strong> charge conservationalso dictate that D-branes and NS5-branes stretch out to infinity; i.e. they can not havedefinite start- and endpoints like e.g. an open string. Only D-branes can have these ifthey end on NS5-branes.5.3 SummaryWe introduced T-duality, and showed that it highly motivates the introduction <strong>of</strong> Dpbranes.Some properties <strong>of</strong> these D-branes were presented, and we analysed whichconfigurations are possible in Type IIA and IIB superstring theory. Finally, we shed adim light on the NS5-brane, and hinted at some <strong>of</strong> its most important features.

Part IIIResearch75

Chapter 6Setting“Alison I’ll drink your wineI’ll wear your clothes when we’re both highAlison, I said we’re sinkingBut she laughs, and tells me its just fineI guess she’s out there, somewhere...”SlowdiveAlisonThe research carried out in this thesis is based on part <strong>of</strong> Raphael Benichou’s doctoralthesis, [3]. The aim is to remake a computation which is carried out using a twodimensionalconformal field theory in said document, using the space-time based <strong>DBI</strong>action, and check to which extend both methods give rise to the same results.6.1 General settingThe setting we consider is that <strong>of</strong> a number k <strong>of</strong> NS5-branes in flat space-time, extendingin dimensions x µ=0,...,5 , equally spread on a circle <strong>of</strong> radius ρ 0 in the (x 6 , x 7 ) plane, andsituated at the origin in the (x 8 , x 9 ) plane (cfr. Fig. 6.1). We call this an NS5-ring.x 8,9x 7x 6ρ 0NS5-braneFigure 6.1: Schematic representation <strong>of</strong> the general background setting.We consider a D4-brane, extending in dimensions x 0,1,2,3,6 (see Table 6.1). In dimensionsx 0,1,2,3 it does not feel the presence <strong>of</strong> the NS5-branes, in the x 6 direction however,76

CHAPTER 6. SETTING 77it does. This should be understood as follows. Recall that D- and NS5-branes are BPSobjects. However, this does not mean that we can put them together in an arbitrary wayand expect the result to also be BPS. Such BPS preserving configurations do neverthelessexist for these two objects. This implies that they will configure themselves in sucha way that the sum <strong>of</strong> the gravitational attraction between both objects (attractive)and the force due to the exchange <strong>of</strong> closed strings (repulsive) cancels. If we were toplace a D4-brane in the vicinity <strong>of</strong> the NS5-ring that deviates from such an equilibriumconfiguration, the D4-brane will move due to the non-cancelling <strong>of</strong> the sum <strong>of</strong> mentionedforces till the configuration becomes stable. Once such a stable configuration is reached,it will remain stable forever. As such, we can basically place our D4-brane in any suchequilibrium position, and rest assured that it will stay there. The NS5-branes curve thespatial dimensions normal to its world volume. Hence, the presence <strong>of</strong> the NS5-branesis felt by the D4-brane through the curvature in the x 6 direction. What makes theD4-brane particularly interesting in this setup, is that four <strong>of</strong> its space-time directionscorrespond to a four-dimensional Minkowski space-time. For, and only for bound states,the x 6 direction is “invisible.” A word <strong>of</strong> explanation is in order.Dim: 0 1 2 3 4 5 6 7 8 9NS5 × × × × × ×D4 × × × × ×Table 6.1: Dimensional overlap between D4- and NS5-branes.6.2 What’s in a title?The title <strong>of</strong> this dissertation reads“<strong>DBI</strong> <strong>Analysis</strong> <strong>of</strong> <strong>Open</strong> <strong>String</strong> <strong>Bound</strong> <strong>States</strong> on Non-compact D-branes.”In Part I <strong>of</strong> this document, the concepts <strong>of</strong> open strings, D-branes and the <strong>DBI</strong> actionhave all been reviewed. There remain two concepts that have not been mentioned yet:what do we understand under bound states, and what are non-compact D-branes? It isalso worth sheding a bit <strong>of</strong> extra light on the <strong>DBI</strong> approach.6.2.1 Compact vs. non-compact D-branesWe have seen that D-branes, when present in some direction, are forced to stretch out toinfinity in that direction due to arguments related to charge conservation. There is oneexception to the rule, and that is when D-branes touch NS5-branes. We can considertwo cases: one where a D-brane “hits” only one NS5-brane, and one where it “hits” twoNS5-branes. We provide “hits” with quotation marks, because as mentioned earlier,these configurations are in fact static. Nevertheless, we will assume it to be dynamicjust for the sake <strong>of</strong> argument.

CHAPTER 6. SETTING 78If we imagine a D1-brane stretching out in x 6 approaching and ultimately hitting anNS5-brane, it will be cut into two pieces, namely two semi-compact D1-branes. Theyare semi-compact, because on one side they stretch out to infinity, while on the otherthey end on the NS5-brane.If we imagine a D1-brane stretching out in x 6 hitting two NS5-branes, it will be cutinto three pieces, namely two semi-compact D1-branes and one compact D1-brane, i.e.a D-brane that ends on both sides on an NS5-brane.Both these possibilities are illustrated in Fig. 6.2. A non-compact D-brane is thussimply a brane that extends to infinity on both sides.D1-braneD1-braneCompactD1NS5-ringSemi-CompactD1Semi-compactD1NS5-ringSemi-compactD1Figure 6.2: Graphical illustration <strong>of</strong> (semi)-compact D1-branes in the presence <strong>of</strong> anNS5-ring.6.2.2 <strong>Bound</strong> statesIt has been mentioned before that open strings can be regarded as excitations <strong>of</strong> D-branes. Such an open string satisfies Dirichlet boundary conditions in the directionsnormal to the brane, meaning in concreto that the endpoints <strong>of</strong> such a string are attachedto the world volume <strong>of</strong> the D-brane. But as long as they remain attached, they are freeto move on it as they please.In the presence <strong>of</strong> an NS5-ring, things change. Focusing again on a D1-branestretched in the x 6 direction, an open string attached to the brane can in principlemove freely along the brane. When the D1-brane is located close enough to the NS5-ring however, something radical happens: some excitations close to the NS5-ring remainbound, i.e. they cannot move anymore! Such a state is exactly what is called a boundstate. This is illustrated in Fig. 6.3.6.2.3 <strong>DBI</strong> approachThe Dirac-Born-Infeld action is said to be a low-energy action. What does this entail?For one, the <strong>DBI</strong> action is constructed entirely out <strong>of</strong> the massless fields present in the((NS,NS)-sector <strong>of</strong> the open) string spectrum. In other words, the <strong>DBI</strong> action onlyconsiders the lowest energy excitations <strong>of</strong> the open string. As such, when we vary the<strong>DBI</strong> action to obtain the excitations <strong>of</strong> the D-brane it describes, and hence the spectrum

CHAPTER 6. SETTING 79<strong>of</strong> the open strings living on it, we only look at low-energetic particles, meaning thelowest excitation states. In concreto, these are the scalar and vector particles.Also, the <strong>DBI</strong> action is only valid if the string coupling is weak. If this is not thecase, one should consider extra terms in a perturbative g s expansion.6.3 Geometrical settingLet us now return to our ten-dimensional Minkowski space-time, with a ring <strong>of</strong> NS5-branes extending in x µ=0,...,5 . The presence <strong>of</strong> these NS5-branes makes the flat spacetimecurved, resulting in the following metric (which is valid also if the NS5-branes arenot spread on a circle):⎧⎪⎨ ds 2 = η µν dx µ dx ν + Hδ ij dx i dx je⎪⎩2Φ = e 2Φ 0H(6.1)= ⋆ 4 dHH (3)with i, j ∈ {6, 7, 8, 9}, H defined as a function <strong>of</strong> the positions x i a <strong>of</strong> the NS5-branes asH(x i ) = 1 +k∑ α ′|x i − x i 2, (6.2)a|a=1and ⋆ 4 the Hodge dual in x i=6,...,9 , making H (3) the three-from field strength <strong>of</strong> theKalb-Ramod field. Note that the NS5-branes behave like black holes. By specializingto the case <strong>of</strong> NS5-branes spread on a circle, introducing the coordinates{(x 6 , x 7 ) = ρ 0 chr sinθ (cos ψ, sinψ),(x 8 , x 9 (6.3)) = ρ 0 shr cos θ (cos φ,sinφ) ,D1-branex 6x 7 x xboundstatefreestatex 7D1-braneCompact D1x 7 x xx 6NS5-ringNS5-ringx 6NS5Figure 6.3: Graphical illustration <strong>of</strong> bound states. When the D-brane is far enough, theexcitations are free (left). When the brane is close enough, they become bound (mid). Ifthe brane gets even closer, it gets cut in three pieces, one <strong>of</strong> which is a compact D-branefixed between two NS5-branes. Also these states are bound (right).

CHAPTER 6. SETTING 80with r ∈Ê+, 0 ≤ θ ≤ π 2and 0 ≤ φ, ψ ≤ 2π, considering the double scaling limitg s ,ρ 0√α ′ρ 0g s√α ′−→ 0, (6.4)= constant, (6.5)and performing a T-duality along the angular φ direction, one is left with the geometry⎧[ ( ) dω 2 ( ) ]dω2⎪⎨ ds 2 = dx µ dx µ + α ′ k dr 2 + coth 2 k + dψ + dθ 2 + tan 2 θ ,k(6.6)⎪⎩ e 2Φ = g2 eff 1k sh 2 r cos 2 θ ,with k the number <strong>of</strong> NS5-branes, ω the T-dual coordinate to φ, and g eff given by√kα ′g eff = e Φ 0. (6.7)ρ 0Note that the Kalb-Ramond field has vanished. Further note that although we nowfind ourselves with three compact dimensions (θ, φ and ψ), we did not compactify anydimension. All we did was introduce a change <strong>of</strong> coordinates, just like one could reachany point on the flat (x, y)-plane by specifying an angle and a radius. By introducingsuch a compact coordinate, one does not compactify a dimension.It is worth pauzing an extra second at this, since this is exactly what sets this approachapart from the Kaluza-Klein-like approach where one does compactify a dimensionin order to make it invisible (or more precisely: make the physics in that dimensiondecouple from the physics in the other dimensions). We do not compactify, but considera setting in which states appear that are fixed in some dimension, limiting their freedom<strong>of</strong> movement to the remaining dimensions, thereby from a physical point <strong>of</strong> view makingthe “static” dimension “invisible.”The bracketed part <strong>of</strong> the geometry described by Eq. 6.6 can be decomposed as thedirect product <strong>of</strong> two two-dimensional geometries. One is the bell,⎧⎨ds 2 = α ′ k [ dθ 2 + tan 2 θ dω 2] ,⎩e 2Φ = g2 effcos 2 θ , (6.8)and the other is the trumpet,⎧⎨ds 2 = α ′ k [ dr 2 + coth 2 r dψ 2] ,⎩e 2Φ = e2Φ 0sh 2 r . (6.9)We will not be concerned by the six-dimensional Minkowski and two-dimensional bellgeometries, as the physics in these geometries is already quite well understood. Weinstead focus on what happens in the trumpet.

CHAPTER 6. SETTING 81It was mentioned above that compact D-branes coined between two NS5-branes alsoexhibit bound states. These bound states are massless. Contrary to this, it has beenshown in [3] that the spectrum <strong>of</strong> the bound states appearing on non-compact D-branesin the close vicinity <strong>of</strong> the NS5-ring have a small mass. Recalling that these statesare bound in the x 6 direction, they can only move in the x 0,1,2,3 directions anymore,which correspond exactly to a 4D-Minkowski space-time. It has been proven, still in [3],that these massive bound states correspond to massive gauge bosons in a 4D Minkowskispace-time. This is very interesting, because it allows for yet another building block <strong>of</strong>the Standard Model to be recreated in string theory. Even though it is not yet fullyunderstood how to use this piece <strong>of</strong> the Standard Model puzzle, at least it is sure to bepresent. This result was shown using exact computations in a conformal field theoreticalapproach. The advantage <strong>of</strong> this approach is that it is exact. The drawback however isthat such a conformal field theoretical formulation is only known for a limited number<strong>of</strong> setups, and in particular it is not known in the region far away from the NS5-ring.On the other hand, the <strong>DBI</strong> approach is easily generalizable to this region, but is onlyvalid when the space-time curvature is small at the string length scale. Hence, it isinteresting to find out to which extend both methods agree in the region where they areboth applicable, so as to be able to judge if whether or not the approximate formulationis exact enough to be useful in the far-away region.Returning to the factorized space-time, remark that since we assumed our D4-braneto live in x 0,1,2,3,6 , it also gets factorized, namely into a D3-brane living in x 0,1,2,3 , and aD1-brane living in the trumpet geometry. However, we are faced with the problem thatthe trumpet geometry explodes as r −→ 0 (see Eq. 6.9 and Fig. 6.4). The problem liesin the fact that, as can be seen from Eq. 6.9, as r approaches 0, e Φ −→ ∞, indicatingthat we can no longer assume the coupling to be weak. As such, if we want to use the<strong>DBI</strong> action, we should consider extra terms in a perturbative g s expansion.Figure 6.4: Graphical representation <strong>of</strong> the trumpet (l) and cigar (r) geometries.

CHAPTER 6. SETTING 82But we can perform a T-duality on the circular ψ dimension <strong>of</strong> the trumpet, resultingin the cigar geometry (see Fig. 6.4), and the change <strong>of</strong> our D4-brane to a D5-brane.In factorized terms, our D3-“Minkowski”-brane remains unchainged, but the D1-branein the trumpet geometry now becomes a D2-brane covering the entire cigar geometry.This geometry is given by⎧⎨ds 2 = α ′ k [ dρ 2 + tanh 2 ρ dτ 2] ,⎩e 2Φ = e2Φ 0ch 2 ρ . (6.10)As far as the presence <strong>of</strong> the NS5-branes goes in these geometries, on the trumpet onecan consider them to be at infinity on the exploding plane at r = 0, and at the originon the cigar.6.4 RoadmapWe have now come to a point where we managed to isolate our test subject in a geometrywe know how to handle. So how should we proceed from here on?We cannot simply write down the <strong>DBI</strong> action for a D-brane in some geometry, aD2-brane in the cigar geometry in our specific case, and start varying around. Whatwe first need to do is to find the classical solution <strong>of</strong> the D-brane in this geometry.The classical solution is what one finds when one varies the unperturbed <strong>DBI</strong> D-braneaction with respect to the coordinates or fields one ultimately wants to vary, or moreprecisely can vary. This solution effectively extremizes the <strong>DBI</strong> action, thus constitutingthe equilibrium solution around which we will want to perturb. As a specific example,consider a D1-brane in the cigar and trumpet geometries. As will be shown in the nextchapter, the classical solutions for these configurations are given bytrumpet:cigar:sin(ψ − ψ 0 ) = Cchρ , (6.11)sin (τ − τ 0 ) = Cshρ . (6.12)Minimizing the energy <strong>of</strong> the D-brane in these configurations, these equations also tell ushow the brane will position itself in these geometries, i.e. they represent the embeddings<strong>of</strong> the D-brane in the geometry. In these embeddings, we see the appearance <strong>of</strong> twointegration constants, C and ψ 0 and τ 0 for the trumpet and cigar respectively. ψ 0 andτ 0 are angles with no physical interpretation; they simply define an <strong>of</strong>fset compared tothe origin in that direction, which amounts to rotating the brane around geometriesthat are symmetrical in that direction, hence not changing the physics. The constant Chowever has a very distinct interpretation: it can be considered as a measure <strong>of</strong> distance<strong>of</strong> the D1-brane to the NS5-ring, as is shown in Figure 6.5.Once we have found the classical solution, we can start thinking about varying theaction. For this, one simply introduces perturbations <strong>of</strong> coordinates and/or fields (e.g.the Maxwell field living on the D-brane). These perturbations will enter the action

CHAPTER 6. SETTING 83through the induced metric, Kalb-Ramond field and Maxwell field strength, these lasttwo <strong>of</strong> course only if present. Specifically, given the general pullback form,T αβ = ∂Xµ∂X α ∂X ν∂X β T µν, (6.13)with T µν some arbitrary tensor, partial derivatives <strong>of</strong> the perturbations will appear. Assuch, when one computes the determinant <strong>of</strong> the resulting matrix (recall the general√−gαβ + B αβ + 2πα ′ F αβ term in the <strong>DBI</strong> action), and expands the square root, oneis typically left with a number <strong>of</strong> zero, first and second order partial derivatives <strong>of</strong> theexpansions. By computing the equations <strong>of</strong> motion <strong>of</strong> this Lagrangian density, oneobtains a differential equation for the perturbations. By solving this equation, one cananalyse the spectrum <strong>of</strong> the perturbations.Summarized in short, to obtain the spectrum <strong>of</strong> the perturbations <strong>of</strong> a D-brane usingthe <strong>DBI</strong> action, and hence <strong>of</strong> the low-energy excitations <strong>of</strong> the D-brane, one shouldfirst write down the <strong>DBI</strong> action for the D-brane in the geometry under consideration,compute the classical solution, introduce perturbations, compute the equations <strong>of</strong> motion<strong>of</strong> these perturbations, and combine them into a differential equation. The solutions <strong>of</strong>this differential equation tells you the spectrum <strong>of</strong> the perturbations.6.5 How to recognize bound statesThe last question left to answer is how to known whether or not the spectrum containsbound states.The crucial point is that in order to solve the equations <strong>of</strong> motion for the perturbations,one always proposes a factorized solution containing a quantum mechanical e iEtterm to remove the always present ∂ 2 t term. This term is always present, because theperturbations are always considered to be time dependent, which by virtue <strong>of</strong> the pullbackmechanism amounts to such a second time derivative. This introduces the energy<strong>of</strong> the fluctuations in the differential equation.Using E 2 = m 2 + p 2 , we can relate this energy to the mass <strong>of</strong> the particles. If onemanages to solve the differential equations resulting from the equations <strong>of</strong> motion, onecan also derive which values <strong>of</strong> E 2 are allowed by the solution. A priori, if nothingbounds the particles, this energy is arbitrary since nothing bounds their impulse, i.e.p ∈ [0, +∞[, resulting in a continuous spectrum. However, for a bound state p = 0, andso the small mass (E 2 = m 2 ) <strong>of</strong> the particle will reveal itself through a discrete value <strong>of</strong>the energy laying below the continuum spectrum.

CHAPTER 6. SETTING 84Figure 6.5: D1-brane embeddings in trumpet (top-left: C = 0.1, mid-left: C = 1,bottom-left: C = 5) and cigar (top-right: C = 0.1, mid-right: C = 1, bottom-right:C = 5) geometries.

Chapter 7A First Example“Le temps n’apprécie rien qui se fait sans lui...”Lambic Brewery, BrusselsWritten on a piece <strong>of</strong> wood patientlyhanging in the fermentation chamber.As a way <strong>of</strong> getting acquainted with the problem at hand, a good place to start is toretrace the steps needed in order to obtain the results reported in [6]. In this article,a calculation is carried out in section 4 ressembling the one we will ultimately want tomake. The problem at hand is to find the spectrum <strong>of</strong> a D1-brane living on the cigartarget space using the small fluctuation <strong>DBI</strong> analysis. We will also perform the samecomputation for a D1-brane living on the trumpet, a case <strong>of</strong> interest since the D2-braneon the cigar we want to analyse came about by T-dualizing a D1-brane on the trumpet.As was mentioned earlier, we should be careful when interpreting the results <strong>of</strong> thiscomputation, since the exploding <strong>of</strong> the Φ field near the trumpet origin alerts us thatwe should also take into account extra perturbative terms. Nevertheless, despite this,going from the cigar to the trumpet is so easy, that this result will constitute a firstindication whether or not we are on the right track (<strong>of</strong> finding bound states, that is).In all the computations in this and the next chapter, we set α ′ = 1.7.1 D1-brane on the cigar7.1.1 Metric and actionThe cigar metric is given byds 2 = (dρ) 2 + (tanh ρ) 2 dτ 2 ; e −Φ = e −Φ 0chρ. (7.1)85

CHAPTER 7. A FIRST EXAMPLE 86The <strong>DBI</strong> action for a D1-brane thus becomes 1 (r is the coordinate on the D-brane)∫S D1 = −T 1 dre −Φ√ ∫detg αβ = −T 1 e −Φ 0drchρ √ det g αβ . (7.2)Since there is only one coordinate, we find the following for the induced metric:g αβ = g rr =( ) ∂ρ 2 ( ) ∂τ 2+ tanh 2 ρ . (7.3)∂r∂rDenoting ∂ r X = X ′ , our action takes on the form∫ √S D1 = −T 1 e −Φ 0dr ch 2 ρ (ρ ′ ) 2 + sh 2 ρ (τ ′ ) 2 . (7.4)Using reparameterization invariance to fixr = shρ, (7.5)we find thatS D1 = −T 1 e −Φ 0∫√dr 1 + r 2 (τ ′ ) 2 . (7.6)7.1.2 Classical solutionThe equations <strong>of</strong> motion for r become( ) ∂L0 = ∂ r∂ (∂ r τ)[= ∂ r − 1 2r 2 τ ′ ]√2 1 + r 2 τ ′= r2 τ ′′ + 2rτ ′√− 1 r 2 τ ′ ( 2r(τ ′ ) 2 + 2r 2 τ ′ τ ′′)1 + r 2 (τ ′ ) 2 2(1 + r 2 (τ ′ ) 2) 3/20 = rτ ′′ + 2τ ′ + r 2 ( τ ′) 3 . (7.7)This can be verified by the relation (recall that r = shρ)sin (τ − τ 0 ) = Cshρ , (7.8)with τ 0 and C constant. This gives the embedding <strong>of</strong> τ, and agrees with the D1 cig entryin Table 2, p.4 in the article.1 Note that since at the moment we do not consider a timelike coordinate on the brane, we takepdet gαβ instead <strong>of</strong> p − det g αβ .

CHAPTER 7. A FIRST EXAMPLE 877.1.3 Adding perturbationsThe next step is to add perturbations. For this, we need to introduce time. We do thisby modifying the metric,ds 2 = − 1 k dt2 + dρ 2 + th 2 ρ dτ 2 , (7.9)in which k is the number <strong>of</strong> NS5-branes spread out on the circle. Instead <strong>of</strong> scaling thetimelike dimension with k −1 we could also have scaled (all) the spacelike dimensionswith k as was done in Eq. 6.6, and one founds both methods in the literature. In terms<strong>of</strong> r this becomes (we also include the restatement <strong>of</strong> e −Φ )⎧⎨ds 2 = − 1 k dt2 + dr21 + r⎩2 + r21 + r 2dτ2 ,(7.10)e −Φ = e 0√ −Φ 1 + r 2 .Now we consider perturbations <strong>of</strong> τ by stating that( ) Cτ = arcsin + δτ (r, t). (7.11)rHence, we see that the components <strong>of</strong> the induced metric becomeUsing∂ r τ =(∂ r τ) 2 =g rt = g tr =g tt = −1k +r21 + r 2 (∂ tδτ) 2 , (7.12)g rr = 11 + r 2 + r21 + r 2 (∂ rτ) 2 , (7.13)r21 + r 2 (∂ tδτ)(∂ r τ). (7.14)−Cr 2 √1 − C2r 2 + (∂ r δτ), (7.15)C 2( ) − 2C √(∂ rδτ)+ (∂ r δτ) 2 , (7.16)r 4 1 − C2r 2 r 2 1 − C2r 2one finds that⎡g αβ =⎢⎣⎡r 2⎣1 + r 2−1k +⎡r21 + r 2 (∂ tδτ) 2 r 2⎣1 + r 2⎤−C√ + (∂ r δτ)r 2 1 − C2r 2⎦(∂ t δτ)⎡1⎣ 11 + r 2 1 − C2−C√ + (∂ r δτ)r 2 1 − C2r 2⎤⎦(∂ t δτ)+ r 2 (∂ r δτ) 2 − 2C √(∂ rδτ)⎦r 2 1 − C2(7.17)r 2 ⎤⎤.⎥⎦

CHAPTER 7. A FIRST EXAMPLE 88From this, one computes that up to second order− ( 1 + r 2) ( ) √1det g αβ = ( )[1 + r 2 1 − C2k 1 − C2r 2 (∂ r δτ) 2 − 2C 1 − C2r 2 (∂ rδτ)r 2 ( ) ]−kr21 + r 2 1 − C2r 2 (∂ t δτ) 2 + O (3). (7.18)This finally allows us to compute the Lagrangian density:√L <strong>DBI</strong> = −T 1 e −Φ 0− (1 + r 2 ) detg αβ[)≈ −T 1 e −Φ 10√ ( ) 1 +(1 r2− C22 r 2 (∂ r δτ) 2k 1 − C2r 2kr 2 ) √−(12 (1 + r 2 − C2) r 2 (∂ t δτ) 2 − C 1 − C2r 2 (∂ rδτ)− 1 ) ) ](4C(1 2 − C28 r 2 (∂ r δτ) 2 + O (3)[L <strong>DBI</strong> = − T 1e −Φ 01√ √ − C (∂ r δτ)k 1 − C2r√2kr 2) 3/2]−2 (1 + r 2 1 − C2) r 2 (∂ tδτ) 2 +(1 r2− C22 r 2 (∂ r δτ) 2 . (7.19)1This result agrees with Eq. (5) in the original paper, except for an overal factor <strong>of</strong> √kwhich has been forgotten in the article, and a minus sign in front <strong>of</strong> the first order termwhich is due to the fact that we used τ = arcsin ( ) (Cr +δτ instead <strong>of</strong> τ = arccos C)r +δτwhich was used in the article. This sign is <strong>of</strong> no importance, as the linear term doesnot contribute to the equations <strong>of</strong> motion. Also note a typo in the article in the term∼ (∂ r δτ) 2 , where the author wrote ( √ ) 3/2 . . . instead <strong>of</strong> (. . .) 3/2 .

CHAPTER 7. A FIRST EXAMPLE 897.1.4 Perturbed equations <strong>of</strong> motionWe proceed to computing the equations <strong>of</strong> motion for the perturbation δτ:∂L0 = ∂ µ∂ (∂ µ δτ) = ∂ ∂Lt∂ (∂ t δτ) + ∂ ∂Lr∂ (∂ r δτ)⎡ √ ⎤ [= ∂ t⎣ −kr2 1 − C2( ) 3/2]r 21 + r 2 (∂ t δτ) ⎦ + ∂ r r 2 1 − C2r 2 (∂ r δτ)= − kr21 + r 2 √1 − C2+(∂2r 2 t δτ ) + 2r( ) 3/2) (r 2 1 − C2 (∂2r 2 r δτ )0 = − kr (∂21 + r 2 t δτ ) )+(2 + C2r 2 (∂ r δτ) + r[) 3/2 (1 − C2r 2 + 3C2r√1 − C2r 2 ](∂ r δτ)(1 − C2r 2 ) (∂2r δτ ) . (7.20)In order to eliminate the time derivative, we fill in δτ = X (r)e iEt , giving us)0 = r(1 − C2 (∂2r 2 r X ) )+(2 + C2r 2 (∂ r X) + krE21 + r2X. (7.21)7.1.5 Solutions <strong>of</strong> the PDETo find the solutions to the differential equation expressed in Eq. 7.21, we will transformit into a hypergeometric equation, which is a second order partial differential equationobeying the general form0 = z (1 − z) ∂ 2 zX + ((c − (a + b + 1) z)∂ z X − abX, (7.22)with a, b and c some arbitrary coefficients. The solutions to the hypergeometric equationcan be expressed in terms <strong>of</strong> the hypergeometric functions. These can be found e.g. in[1] Ch. 2 and [8] Ch. 9.To be able to rewrite Eq. 7.21 into a hypergeometric equation, we introduce thechange <strong>of</strong> variables:r 2 = w, (7.23)w = ( 1 + C 2) z − 1. (7.24)The first substition makes for the following change in derivatives:∂ 2∂r2= ∂w∂r∂∂r = ∂w ∂∂r ∂w = 2√ w∂ w ,[ ]∂ ∂w ∂= 4w∂w 2 + 2∂ w . (7.25)∂w ∂r ∂w

CHAPTER 7. A FIRST EXAMPLE 90Using this, we find0 = k√ wE 2 ((1 + w) X + 2 + C2w)2 √ w∂ w X + √ w(= kE2(1 + w) X + 4 + 2C2w + 2 − 2C2w)∂ w X +) (1 − C2 [4w∂2w X + 2∂ w X ]w(1 − C2w)4w∂ 2 wX= kE2(1 + w) X + 6∂ wX + 4 ( w − C 2) ∂ 2 wX. (7.26)The second substitution results in0 = 4((1 + C2 ) z − 1 − C 2)(1 + C 2 ) 2 ∂ 2 zX +Multiplying through with −z ( 1 + C 2) /4 we get6(1 + C 2 ) ∂ zX +kE 2((1 + C 2 X. (7.27))z)0 = z(1 − z)∂zX 2 − 3 2 z∂ zX − kE2 X, (7.28)4which is the hypergeometric equation we were looking for. The coefficients are given bya = 1 (1 + √ )1 + 4kE42 ; b = 1 (1 − √ )1 + 4kE42 ; c = 0.These coefficients obey c = 1 − m, with m ∈Æ, and a and b /∈ {0, −1, −2, . . .,1 − m},which according to e.g. case 5 on p.1012 in [8] means that the general solution is givenbyThis solution is valid foru = z m 2F 1 (a + m, b + m; 1 + m; z) (7.29)( 1= z 2 F 1(5 + √ )1 − 4kE42 , 1 (5 − √ ) )1 − 4kE42 ; 2;z . (7.30)|z| =(1 + r 2 )∣(1 + C 2 ) ∣ < 1.To obtain the solutions for |z| > 1, one has to use hypergeometric identities, whichamounts to expressing a hypergeometric function in terms <strong>of</strong> other hypergeometric functions.Specifically, one can express a 2 F 1 (. . .,z) function in terms <strong>of</strong> two 2 F 1(. . .,z−1 )functions. A look on p. 105 in [1] learns us that our hypergeometric function (Eq. 7.29)is <strong>of</strong> the same form as u 5 in Eq. 17 in the book. Page 108 then tells us that we canexpress this u 5 as a sum <strong>of</strong> u 3 and u 4 , two hypergeometric functions that are indeed <strong>of</strong>the right form 2 F 1(. . .,z−1 ) , namelyu 5 =Γ (2 − c)Γ(b − a)Γ (1 − a)Γ(b + 1 − c) eiπ(1−c) u 3Γ (2 − c)Γ(a − b)Γ (1 − b) Γ (a + 1 − c) eiπ(1−c) u 4 , (7.31)

CHAPTER 7. A FIRST EXAMPLE 91with Γ the Euler gamma function, andu 3 = (−z) −a 2F 1(a, a + 1 − c; a + 1 − b; z−1 ) , (7.32)u 4 = (−z) −b 2F 1(b + 1 − c, b; b + 1 − a; z−1 ) . (7.33)The large r solution to our hypergeometric equation thus becomesu = C 1 (−1) 1 4(5+ √ ( )11+4kE 2 ) 1 + C21 + r 2(1 + √ 1 + 4kE 2 ), 1 4( 1× 2 F 14+ C 2 (−1) 4(3− 1 √ ( )11+4kE 2 ) 1 + C21 + r 2× 2 F 1( 14(5 − √ 1 + 4kE 2 ), 1 44(1+ √ 1+4kE 2 )(5 + √ )1 + 4kE 2 ; 1 + 1 √)1 + 4kE22 ; z −14(1− √ 1+4kE 2 )(1 − √ )1 + 4kE 2 ; 1 − 1 √)1 + 4kE22 ; z −1 ,(7.34)withC 1 =C 2 =( )Γ − 1 2√1 + 4kE 2( ) ( ), (7.35)Γ 34 − 1 4√1 + 4kE 2 Γ 54 − 1 4√1 + 4kE 2( )Γ 12√1 + 4kE 2( ) ( ). (7.36)Γ 34 + 1 4√1 + 4kE 2 Γ 54 + 1 4√1 + 4kE 2This looks nothing less than atrocious, but can be approximated for r −→ ∞ by using2F 1 (a, b; c; 0) = 1, resulting inu = C 1 (−1) 1 4(5+ √ 1+4kE 2 ) ( 1 + C 2)1 4(1+ √ 1+4kE 2 ) r12(1+ √ 1+4kE 2 )+ C 2 (−1) 1 4(3− √ 1+4kE 2 ) ( 1 + C 2)1 4(1− √ 1+4kE 2 ) r12(1− √ 1+4kE 2 ) . (7.37)7.2 D1-brane on the trumpetWe will redo the calculation carried out in the previous section for the trumpet geometry.The computation is very similar, and hence considerably less steps will be shown explicitely.Recall that this setup suffers from bad string coupling behaviour at the origin,and that hence we should be careful when interpreting the <strong>DBI</strong> analysis, as we shouldin fact consider extra perturbative terms. However, because this setup is T-dual to thewell-behaved D2-brane in the cigar, these results nevertheless give a first indication <strong>of</strong>what to expect.

CHAPTER 7. A FIRST EXAMPLE 927.2.1 Metric and actionThe trumpet metric is given byds 2 = (dρ) 2 + (coth ρ) 2 dψ 2 ; e −Φ = e −Φ 0shρ. (7.38)The <strong>DBI</strong> action for a D1-brane thus becomes (r is the coordinate on the D-brane)∫S D1 = −T 1 dr e −Φ√ ∫det g αβ = −T 1 e −Φ 0dr shρ √ det g αβ . (7.39)The induced metric becomesg αβ = g rr =( ) ∂ρ 2 ( ) ∂ψ 2+ coth 2 ρ . (7.40)∂r∂rDenoting ∂ r X = X ′ , our action takes on the form∫ √S D1 = −T 1 e −Φ 0dr sh 2 ρ (ρ ′ ) 2 + ch 2 ρ (ψ ′ ) 2 . (7.41)Using reparameterization invariance to fixr = chρ, (7.42)we find thatS D1 = −T 1 e −Φ 0∫√dr 1 + r 2 (ψ ′ ) 2 . (7.43)7.2.2 Classical solutionThe equations <strong>of</strong> motion for r become( ) ∂L0 = ∂ r∂∂ r ψ= r2 ψ ′′ + 2rψ ′√− 1 r 2 ψ ′ ( 2r(ψ ′ ) 2 + 2r 2 ψ ′ ψ ′′)r 2 − 1 (ψ ′ ) 2 2(r 2 − 1 (ψ ′ ) 2) 3/20 = rψ ′′ + 2ψ ′ + r 2 ( ψ ′) 3 . (7.44)This can be verified by the relation (recall that r = chρ)with ψ 0 and C constant. This gives the embedding <strong>of</strong> ψ.sin(ψ − ψ 0 ) = Cchρ , (7.45)

CHAPTER 7. A FIRST EXAMPLE 937.2.3 Adding perturbationsWe again need to introduce time in order to be able to perturb the system. Just as forthe cigar, we do this by modifying the metric tods 2 = − 1 k dt2 + dρ 2 + coth 2 ρ dψ 2 , (7.46)which in terms <strong>of</strong> r becomes⎧⎨ds 2= − 1 k dt2 + dr2r⎩2 − 1 +e −Φ = e 0√ −Φ r 2 − 1.r2r 2 − 1 dψ2 ,(7.47)The perturbations are given byψ = arcsin( Cr)+ δψ (r, t), (7.48)resulting in the induced perturbed metric⎡g αβ =⎢⎣⎡r 2⎣r 2 − 1−1k +r2r 2 − 1 (∂ tδψ) 2 r 2−C√ + (∂ r δψ)r 2 1 − C2r 2Up to second order, one finds thatkr2r 2 − 1⎤⎦(∂ t δψ)− ( r 2 − 1 ) 1det g αβ = ( )k 1 − C2r 2−(1 − C2⎡⎣r 2 − 1⎡1r 2 − 1−C√ + (∂ r δψ)r 2 1 − C2r 2⎣ 11 − C2⎤⎦(∂ t δψ)+ r 2 (∂ r δψ) 2 − 2C √(∂ rδψ)⎦r 2 1 − C2(7.49)( ) √[1 + r 2 1 − C2r 2 (∂ r δψ) 2 − 2C 1 − C2r 2 (∂ rδψ)r 2 )(∂ t δψ) 2 ]+ O (3). (7.50)r 2 ⎤⎤.⎥⎦The Lagrangian density thus becomes√L <strong>DBI</strong> = −T 1 e −Φ 0− (r 2 − 1) detg αβ[= − T 1e −Φ 01√ √ − C (∂ r δψ)k 1 − C2r√2kr 2) 3/2]−2 (r 2 1 − C2− 1) r 2 (∂ tδψ) 2 +(1 r2− C22 r 2 (∂ r δψ) 2 . (7.51)

CHAPTER 7. A FIRST EXAMPLE 947.2.4 Perturbed equations <strong>of</strong> motionWe again proceed to computing the equations <strong>of</strong> motion for the perturbation δψ,∂L0 = ∂ µ∂ (∂ µ δψ) = ∂ ∂Lt∂ (∂ t δψ) + ∂ ∂Lr∂ (∂ r δψ)⎡ √ ⎤ [= ∂ t⎣ −kr2 1 − C2( ) 3/2]r 2r 2 (∂ t δψ) ⎦ + ∂ r r 2 1 − C2− 1r 2 (∂ r δψ)= − kr (∂2r 2 − 1 t δψ ) )+(2 + C2r 2 (∂ r δψ) + r(1 − C2r 2 ) (∂2r δψ ) . (7.52)Filling in δψ = X (r)e iEt , this finally give us the following differential equation:)0 = r(1 − C2 (∂2r 2 r X ) )+(2 + C2r 2 (∂ r X) + krE2r 2 X. (7.53)− 17.2.5 Solutions <strong>of</strong> the PDETo transform Eq. 7.53 into a hypergeometric equation, the needed change <strong>of</strong> variablesare almost the same as for the cigar, namelyThe first substition amounts toThe second substitution results inr 2 = w, (7.54)w = ( C 2 − 1 ) z + 1. (7.55)0 = kE2(w − 1) X + 6∂ wX + 4 ( w − C 2) ∂ 2 wX. (7.56)(z − 1)0 = 4(C 2 − 1) ∂2 zX +6(C 2 − 1) ∂ zX +Multiplying through with −z ( C 2 − 1 ) /4 we getThe coefficients are still given bya = 1 (1 + √ )1 + 4kE42resulting in the same general solution,kE 2((C 2 X. (7.57)− 1)z)0 = z(1 − z)∂zX 2 − 3 2 z∂ zX − kE2 X. (7.58)4; b = 1 4(1 − √ 1 + 4kE 2 ); c = 0,u = z m 2F 1 (a + m, b + m; 1 + m; z)( 1= z 2 F 1(5 + √ )1 + 4kE42 , 1 (5 − √ ) )1 + 4kE42 ; 2;z . (7.59)

CHAPTER 7. A FIRST EXAMPLE 95This time however, the solution is valid for|z| =(r 2 − 1)∣(C 2 − 1) ∣ < 1.The analytic continuation, obtained in the same way as for the cigar, is given byu = C 1 (−1) 1 4(5+ √ 1+4kE 2 ) ( C 2 − 1 )1 4(1+ √ 1+4kE 2 ) r12(1+ √ 1+4kE 2 )+ C 2 (−1) 1 4(3− √ 1+4kE 2 ) ( C 2 − 1 )1 4(1− √ 1+4kE 2 ) r12(1− √ 1+4kE 2 ) , (7.60)withC 1 =C 2 =( )Γ − 1 2√1 + 4kE 2( ) ( ), (7.61)Γ 34 − 1 4√1 + 4kE 2 Γ 54 − 1 4√1 + 4kE 2( )Γ 12√1 + 4kE 2( ) ( ). (7.62)Γ 34 + 1 4√1 + 4kE 2 Γ 54 + 1 4√1 + 4kE 2

Chapter 8Computations“Trilingue! hablo espanol, franceset le language des bêtesJ’accepte d’utiliser la manière forte quand elle s’y prête”La Cliqua (Rocca)Comme Une SarbacaneThis chapter will go through all necessary motions to compute the equations <strong>of</strong> motionfor the fluctuations <strong>of</strong> a D2-brane in a cigar geometry. Already giving away a majorspoiler, we will stumble on a problem for which no apparent solution seems to be athand.8.1 Metric and field strengthThe geometry <strong>of</strong> the cigar is given by{ds 2 = − 1 k dt2 + dρ 2 + tanh 2 ρ dτ 2 ,e −Φ= e −Φ 0chρ.(8.1)The metric thus isThe D2-brane action is given byS D2 = −T 2 e Φ 0⎡⎤−10 0kg µν = ⎢⎣0 1 0 ⎥⎦ . (8.2)0 0 tanh 2 ρ∫√dαdβchρ − det(g αβ + 2πF αβ ), (8.3)96

CHAPTER 8. COMPUTATIONS 97and the induced metric g αβ equalsg αβ =⎡⎢⎣⎤−k −1 0 00 (∂ α ρ) 2 + th 2 ρ (∂ α τ) 2 ∂ α ρ∂ β ρ + th 2 ρ∂ α τ∂ β τ ⎥⎦ . (8.4)0 ∂ α ρ∂ β ρ + th 2 ρ∂ α τ∂ β τ (∂ β ρ) 2 + th 2 ρ (∂ β τ) 2Using parameterization invariance to set{α= chρ,β = τ,(8.5)we find the induced metric to be⎡−1⎤0 0kg αβ =1⎢0⎣ α 2 0− 1 ⎥α 2 ⎦ , (8.6)− 10 0α 2and the induced field strengthF αβ =⎡⎢⎣0 0 00 0 F ατ0 −F ατ 0⎤⎥⎦ , (8.7)withF ατ =1√α 2 − 1 F ρτ. (8.8)8.2 Classical solutionTo find the embedding <strong>of</strong> the D2-brane, we use a Lagrange multiplier, as shown in [15],§4.2. We need to minimizeS D2 − √ λ (∫ ) ∫ √ ( ) 1 1F − N = A dαdτ αk k√2 α 2 + 4π2 Fατ2 − √ λ ∫Fdαdτ √ ατk α 2 − 1= √ A ∫ [ √1dαdτ + α 2 π 2 Fατ 2 − λ ]F√ ατ, (8.9)k A α 2 − 1with A = T 2 e −Φ 0. We find thatδ (S − λ . . .)δF ατ= A √k∫[4π 2 α 2 F ατdαdτ √ − λ ]δF. (8.10)1 + 4π 2 α 2 F ατ A

CHAPTER 8. COMPUTATIONS 98Hence,Using Eq. 8.8, we are finally left withwhere we have defined C = λ2πA .(4π 2 α 2 ) 2 Fατ 2 = λ2 (1 + 4π 2A 2 α 2 Fατ2 )⇓]4π 2 α 2 Fατ[4π 2 2 α 2 − λ2A 2 = λ2A 2⇓4π 2 α 2 Fατ 2 λ 2=4π 2 α 2 A 2 − λ 2⇓λ2πF ατ =α √ (8.11)4πα 2 A 2 − λ2. 2πF ρτ = C√ α 2 − 1α √ α 2 − C 2 =8.3 Field strength variation√ C tanh ρ(8.12)chρ 2 − C2, Since our D2-brane is spread out over the entire cigar, we cannot perturb it in the ρand τ directions. We can however perturb the Maxwell field that lives on it. Therefore,considerA µ −→ A µ + δα µ (t, ρ, τ). (8.13)The variation <strong>of</strong> the field strength thus becomesDenoting ˜F ατ = F ατ + δF ατ , we find that⎡−1kg αβ + 2π ˜F αβ =⎢⎣− 2π∂ tα ρ√α 2 − 1Up to second order, we find that− det(g αβ + 2π ˜F)αβδF ρτ = ∂ ρ α τ − ∂ τ α ρ = f. (8.14)2π∂ t α ρ√α 2 − 11α 2 − 1−2π∂ t α τ − 2π √α 2 − 1 [F ρτ + f]= 1α 2 k − 4π2 (∂ t α τ ) 2α 2 − 14π 2k (α 2 − 1)2π∂ t α τ2π√α 2 − 1 [F ρτ + f]α 2 − 1α 2⎤. (8.15)⎥⎦− 4π2 (∂ t α ρ ) 2α 2 +[F2ρτ + 2F ρτ f + f 2] . (8.16)

CHAPTER 8. COMPUTATIONS 998.4 Regularized LagrangianBefore varying the action containing the perturbed field strength, and ultimatly computingthe equations <strong>of</strong> motion for the fluctuation, we note that we will again need toinvoke the Lagrange multiplier. As such, we first computeL <strong>DBI</strong> −λf√k (α 2 − 1) . (8.17)To this end, the first thing to do is to expand the Lagrangian density,√L <strong>DBI</strong> = A −α 2 det(g αβ + 2π ˜F)αβ{ 1= Ak − 4π2 α 2α 2 − 1 (∂ tα τ ) 2 − 4π 2 (∂ t α ρ ) 2 + 4π2 α 2k (α 2 − 1) F ρτ2}+ 8π2 α 2k (α 2 − 1) F ρτf + 4π2 α 2k (α 2 − 1) f2= A√(α 2 − 1) + 4π 2 α 2 F 2 ρτk (α 2 − 1){4π 2 α 2 k1 −(α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α τ ) 24π 2 ( α 2 − 1 ) k−(α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α ρ ) 2 8π 2 α 2+(α 2 − 1) + 4π 2 α 2 Fρτ2 F ρτ f4π 2 α 2+(α 2 − 1) + 4π 2 α 2 Fρτ2 f}. 2 (8.18)For the expansion, we find up to second order:√{(α 2 − 1) + 4π 2 α 2 Fρτ2 2π 2 α 2 kL <strong>DBI</strong> ≈ Ak (α 2 1 −− 1) (α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α τ ) 22π 2 ( α 2 − 1 ) k−(α 2 − 1) + 4π 2 α 2 Fρτ2 (∂ t α ρ ) 2 4π 2 α 2+(α 2 − 1) + 4π 2 α 2 Fρτ2 F ρτ f[]2π 2 α 28π 4 α 4 Fρτ2 +(α 2 − 1) + 4π 2 α 2 Fρτ2 − ( )(α 2 − 1) + 4π 2 α 2 Fρτ2 2f}. 2 (8.19)Looking back at Eq. 8.17, we see we only need to consider the term ∼ F ρτ in thisexpansion when computing the effect <strong>of</strong> the Lagrange multiplier. We will want bothterms to annul each other, as otherwise the equations <strong>of</strong> motion will contain constantterms (i.e. without f) we do not want. Anticipating this result, in order not to clutterpages with unreadable derivations, we work out the ∼ F ρτ term when filling in the

CHAPTER 8. COMPUTATIONS 100classical solution Eq. 8.12:O(1)[L <strong>DBI</strong> ] =4π 2 α 2 A√ F ρτ fk (α 2 − 1)√(α 2 − 1) + 4π 2 α 2 Fρτ2A 2πα 2 λ √ α= √ 2 − 1k (α 2 − 1) α √ f4π 2 α 2 A 2 − λ√(α 2( 2 − 1) + λ2 α 2 − 1 )4π 2 α 2 A 2 − λ 2= A √k2πλα√4π 2 α 2 A 2 − λ 2 √4π 2 α 2 A 2 − λ 2√(α 2 − 1)(4π 2 α 2 A 2 − λ 2 ) + λ 2 (α 2 − 1) f= A √k2πλα√(α 2 − 1)4π 2 α 2 A 2f=λf√k (α 2 − 1) . (8.20)We thus indeed find thatL <strong>DBI</strong> −λf√ = 0, (8.21)k (α 2 − 1)as we hoped. Further using the classical solution to simplify the factor in front <strong>of</strong> whatis left <strong>of</strong> our Lagrangian density, we see that√(α 2 − 1) + 4π 2 α 2 F 2 ρτk (α 2 − 1)=√1k + C 2k (α 2 − C 2 ) =α√k (α 2 − C 2 ) , (8.22)leaving us with{αL <strong>DBI</strong> = A √k (α 2 − C 2 ) − 2π2 α √ k (α 2 − C 2 )α 2 (∂ t α τ ) 2 − 2π2√ k (α 2 − C 2 )(∂ t α ρ ) 2− 1α[f 22π 2 α 2 [( α 2 − 1 ) + 4π 2 α 2 F 2 ] ]ρτ − 8π 4 α 4 Fρτ2 + √ [ ]k (α 2 − 1) + 4π 2 α 2 Fρτ2 3/2√ .α 2 − 1(8.23)If one further fills in the classical solution for F ρτ , one finds thatL <strong>DBI</strong> = 2π 2 A √ {k (α 2 − C 2 α)2π 2 [k (α 2 − C 2 )] − αα 2 − 1 (∂ tα τ ) 2 − (∂ tα ρ ) 2α(α 2 − C 2)+k (α 2 − 1)α[(∂ ρ α τ ) 2 − 2 (∂ ρ α τ ) (∂ τ α ρ ) + (∂ τ α ρ ) 2] } . (8.24)

CHAPTER 8. COMPUTATIONS 1018.5 Equations <strong>of</strong> motionFrom the Lagrangian density expressed in Eq. 8.24, we will now compute the equations<strong>of</strong> motion. We start with α ρ .∂ µ∂L∂ (∂ µ α ρ ) = ∂ t( ) ( ∂L ∂L+ ∂ τ∂ (∂ t α ρ )= ∂ t(− 2 (∂ tα ρ )α)∂ (∂ τ α ρ )) ( (2 α 2 − C 2)+ ∂ τk (α 2 − 1)α [(∂ τα ρ ) − (∂ ρ α τ )]= − ( ∂ 2 t α ρ)−(α 2 − C 2)k (α 2 − 1) (∂ τf) . (8.25)The equations <strong>of</strong> motion for α τ are unfortunatly a tad more complicated.( ) ( )∂L ∂L ∂L∂ µ∂ (∂ µ α τ ) = ∂ t + ∂ ρ∂ (∂ t α τ ) ∂ (∂ ρ α τ )( ) [−2α= ∂ tα 2 − 1 (∂ tα τ ) + ∂α(2 α 2∂ρ ∂ − C 2) ]αk (α 2 − 1)α f= − ( ∂ 2 t α τ)+α 2 − C 2kα 2 (∂ ρ f) + −α2 ( α 2 + 1 ) + 3C 2 α 2 − C 2kα 3 (α 2 − 1) 1/2 f. (8.26))From this we conclude that⎧ (( )⎪⎨ ∂2 α 2 − C 2)t α ρ = −k (α 2 − 1) (∂ τf) ,(( )⎪⎩ ∂2 α 2 − C 2) (t α τ =kα 2 (∂ ρ f) − α2 α 2 + 1 ) − C 2 ( 3α 2 − 1 )kα 3 (α 2 − 1) 1/2 f.(8.27)8.6 Road to differential equationWe are looking for the equation <strong>of</strong> motion <strong>of</strong> f. We need to construct this using Eq.8.27. To combine both equations into one single equation for f, we need to derive thefirst equation with respect to τ, the second equation with respect to ρ, and then finallytake the difference <strong>of</strong> these derivations. The first one is again the easiest;(α∂t 2 2 − C 2) ((∂ τ α ρ ) = − ∂2k (α 2 − 1) τ f ) . (8.28)

CHAPTER 8. COMPUTATIONS 102The second one needs some more care, as the factor preceding the partial derivatives <strong>of</strong>f is α-dependent. We see that(α∂t 2 2 − C 2) ((∂ ρ α τ ) = ∂2kα 2 ρ f ) ( − α2 α 2 + 1 ) − C 2 ( 3α 2 − 1 )kα 3 (α 2 − 1) 1/2 (∂ ρ f)+ (∂ ρ f) √ ( α 2α 2 − C 2 )− 1∂ αkα 2Using∂ α(α2 ( α 2 + 1 ) − C 2 ( 3α 2 − 1 )kα 3 (α 2 − 1) 1/2 )− f √ α 2 − 1∂ α(α2 ( α 2 + 1 ) − C 2 ( 3α 2 − 1 )kα 3 (α 2 − 1) 1/2 )∂ α( α 2 − C 2kα 2 )= − α4 + α 2 − 3C 2 α 2 + C 2[(α 2 − 1) α] 2 ,. (8.29)= α4 ( −3 + 6C 2) + α 2 ( 1 − 7C 2) + 3C 2kα 4 (α 2 − 1) 3/2 , (8.30)this amounts to∂ 2 t (∂ ρ α τ ) =(α 2 − C 2) (∂2kα 2 ρ f ) (α 4 + α 2 ( 1 − 5C 2) + 3C 2)−kα 3 (α 2 − 1) 1/2 (∂ ρ f)(3C 2 + α 2 ( 1 − 7C 2) + α 4 ( −3 + 6C 2))−kα 4 (α 2 f. (8.31)− 1)8.7 Differential equationSubtracting Eq. 8.28 from Eq. 8.31, we find that(∂2t f ) (α 2 − C 2) (= ∂2kα 2 ρ f ) (α 4 + α 2 ( 1 − 5C 2) + 3C 2)−kα 3 (α 2 − 1) 1/2 (∂ ρ f)(3C 2 + α 2 ( 1 − 7C 2) + α 4 ( −3 + 6C 2))−kα 4 (α 2 f− 1)(α 2 − C 2) (+ ∂2k (α 2 − 1) τ f ) . (8.32)We still have ρ derivatives working in on f. We replace these by α derivatives by using∂∂ρ = ∂α ∂∂ρ ∂α = (shρ)∂ α = ( α 2 − 1 ) 1/2∂α , (8.33)∂ 2∂ρ 2 = ∂α ( )∂ ∂α ∂= α∂ α + ( α 2 − 1 ) ∂ 2∂ρ ∂α ∂ρ ∂αα, (8.34)

CHAPTER 8. COMPUTATIONS 103which delivers0 = − ( ∂t 2 f ) (α 2 − C 2) (+ ∂2k (α 2 − 1) τ f ) (α4 ( 6C 2 − 3 ) + α 2 ( −7C 2 + 1 ) + 3C 2)−kα 4 (α 2 f− 1)+ 1 [α 5kα 3 − α 4 − α 3 C 2 + ( 5C 2 − 1 ) α 2 − 3C 2] (∂ α f)(α 2 − C 2) ( α 2 − 1 )+kα 2 (∂2α f ) . (8.35)To get a second order partial differential equation in ∂ α , we suppose thatf(t, ρ, τ) = e iEt e iLτ f(ρ), (8.36)which finally delivers the sought after equation <strong>of</strong> motion for f:0 =[kE ( 2 − L2 α 2 − C 2) (k (α 2 − α4 6C 2 − 3 ) + α 2 ( −7C 2 + 1 ) ]+ 3C 2− 1)kα 4 (α 2 f− 1)+ 1kα 3 (α 5 − α 4 − α 3 C 2 + ( 5C 2 − 1 ) α 2 − 3C 2) (∂ α f)(α 2 − C 2) ( α 2 − 1 )kα 2 (∂2α f ) . (8.37)8.8 Hypergeometric equation?We introduce the change <strong>of</strong> coordinatesThe partial derivatives thus becomeThus, we find that[kE ( 2 − L2 ω − C 2)0 =α 2 −→ ω. (8.38)∂∂α = ∂w ∂∂r ∂w = 2√ w∂ w ,∂ 2∂α 2 = ∂w [ ]∂ ∂w ∂= 4w∂w 2 + 2∂ w . (8.39)∂r ∂w ∂r ∂w(− ω2 6C 2 − 3 ) + ω ( −7C 2 + 1 ) ]+ 3C 2ω 2 f(ω − 1)(ω − 1)+ 2 [ω 5/2 − ω 3/2 C 2 + ω ( 4C 2 − 2 ) + 2C 2] (∂ ω f)ω+ 4 ( ω − C 2) (ω − 1) ( ∂ωf 2 ) . (8.40)We can already see that what we did in the previous chapter will not work this time.The problem lies in the abundant presence <strong>of</strong> w in the ∼ f and ∼ ∂ ω f terms. But let us

CHAPTER 8. COMPUTATIONS 104see what comes out anyway. Concentrating on the ∼ ∂ 2 ω term, we see we can put thisterm in the right form by applying the substitutionThe second order term hereby becomesω −→ −z ( 1 − C 2) + 1. (8.41)4 ( ω − C 2) (ω − 1) ( ∂ 2 ωf ) −→ −4 (1 − z)z. (8.42)The only thing left to do is multiply through by −4 −1 . Going through the same motionsfor the other terms, the fully transformed equation becomes (courtesy <strong>of</strong> Mathematica):0 ={ [2 − ( −5C 2 + 5 + kE 2) z + ( C 2 − 1 ) ( 6C 2 − 3 − 2kE 2) z 2 − ( C 2 − 1 ) 2kE 2 z 3+ L 2 (z − 1) ( 1 + ( C 2 − 1 ) z ) ] [2× 4z ( 1 + ( C 2 − 1 ) z ) ] } −12f−{ [− 2 + 6C 2 + 2z − 6C 2 z + 4C 4 z − C 2 ( 1 + ( C 2 − 1 ) z ) 3/2+ ( 1 + ( C 2 − 1 ) z ) ] [5/2× 2 ( C 2 − 1 )( 1 + ( C 2 − 1 ) z ) ] } −1∂ z f+ z (1 − z)∂ 2 zf. (8.43)This is decidely not a hypergeometric equation.

Chapter 9Conclusion & Final Thoughts“I know what I’ve done, but tell me what did I miss?”Queens Of The Stone AgeSick, Sick, SickWe derived the equations <strong>of</strong> motion for the fluctuations <strong>of</strong> a D2-brane in the cigargeometry, and managed to combine them into a second order partial differential equation,Eq. 8.37. We unfortunately found out that we run into problems when we try to rewritethis equation as a hypergeometric equation, preventing us from obtaining the spectrum<strong>of</strong> the D2-brane fluctuations, and concluding the research.The problems arise starting from Eq. 8.27, where the ρ derivative also acts on thefactor in front <strong>of</strong> the fluctuation. This happens a second time right after in Eq. 8.29,resulting in a number <strong>of</strong> α-dependent terms in the zero-order and first-order differentialterms that were absent from the D1-brane case. T-duality however strongly suggeststhat it should nevertheless be possible to rewrite Eq. 8.37 as a hypergeometric equation,although this seems a rather non-trivial task.105

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List <strong>of</strong> AbbreviationsBPSCFTD<strong>DBI</strong>EOMGSGSOKKNNSP.B.’sPDEQCDQFTRRNSSUSYBogomolny-Prasad-SommerfieldConformal Field TheoryDirichletDirac-Born-InfeldEquations Of MotionGreen-SchwarzGliozzi-Scherk-OliveKaluza-KleinNeumannNeveu-SchwarzPoisson BracketsPartial Differential EquationQuantum ChromoDynamicsQuantum Field TheoryRamondRamond-Neveu-SchwarzSUper SYmmetry109

⌈I’m so glad you cameI’m so glad you rememberedTo see how we’re endingOur last dance together. . .⌋[ The Cure (R. Smith) - Last Dance ]