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TACHYON CONDENSATION in STRING FIELD THEORY: the Tachyon Potential in the Conformal field Theory Approach

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TACHYON CONDENSATION in STRING FIELD THEORY: the Tachyon Potential in the Conformal field Theory Approach Katholieke Universiteit Leuven Faculteit der Wetenschappen Instituut voor Theoretische Fysica TACHYON CONDENSATION IN STRING FIELD THEORY: the tachyon potential in the conformal field theory approach Joris Raeymaekers Promotor: Prof. Dr. W. Troost Proefschrift ingediend voor het behalen van de graad van Doctor in de Wetenschappen 2001 ii “It does seem that these physicists go to some effort and expense to state what seems obvious.” William S. Burroughs iii iv Dankwoord Allereerst wil ik mijn promotor, Walter Troost, oprecht bedanken voor zijn begeleiding de afgelopen zes jaar. Eigenlijk begon deze reeds vroeger, toen ik in de eerste licentie onder zijn begeleiding en deze van Urban Studer (die ik langs deze weg ook nog eens wens te bedanken) mocht kennismaken met de fundamenten van de Lorentz-Dirac vergelijking; of hoe een enkele wiskun- dige formule een waaier van fysische verschijnselen kan samenvatten. Het is ongetwijfeld deze kennismaking die mijn fascinatie voor de theoretische natuurkunde blijvend heeft opgewekt. De afgelopen jaren hebben discussies met Walter me talloze keren naar de kern van de zaak gebracht en opnieuw op het juiste spoor gezet. Ook ben ik hem zeer dankbaar voor de grondigheid waarmee hij dit manuscript aan kritische lezing heeft onderworpen: zonder zijn inbreng zouden ongetwijfeld talloze onduidelijkheden, onnauwkeurighe- den en fouten tegen de logica aan mijn aandacht ontsnapt zijn. Ook Toine Van Proeyen wil ik van harte bedanken, voor zijn inzet in het levendig houden van de onderzoeksgroep en voor zijn no-nonsense organi- satorisch talent, ook en vooral op het gebied van de Europese samenwerking, hetgeen ons doctoraatsstudenten ieder jaar opnieuw in staat stelt deel te ne- men aan internationale conferenties en scholen. Ook ben ik Alex Sevrin dankbaar voor een aangename samenwerking, voor zijn nooit aflatende interesse en enthoesiasme, voor het suggereren van in- teressante onderzoeksprojecten evenals, niet te vergeten, mijlpalen van de popgeschiedenis. De resultaten die in deze thesis worden gerapporteerd zouden echter niet tot stand zijn gekomen zonder samenwerking met Pieter-Jan De Smet. Hem wens ik dan ook oprecht te bedanken voor de verhelderende discussies en voor de vele malen dat zijn op wiskundige leest geschoeide zin voor logica orde schiep in mijn verwarring, alsook voor de vele ontspannende conversa- ties over het uitstippelen van goedkope, doch ethisch onverantwoorde Mars- missies en dies meer. Mijn eerste jaren op ‘het zesde’ zouden er niet hetzelfde hebben uitgezien zonder de aanwezigheid van Frederik Denef; hem bedank ik dan ook voor het delen van zijn enorm fysisch inzicht, voor zijn pedagogische uitleg, voor de vele toffe en geestige momenten in Leuven en, de afgelopen jaren, de sappige verhalen over het leven in de Big Apple. Mijn dank gaat ook uit naar Andr´eVerbeure voor de deskundige leiding van het instituut; Robertus Potting, Mark Fannes en Raymond Gastmans voor discussies en aanmerkingen ter verbetering van deze tekst; Jos Rogiers en Raf Dekeyser voor de vlotte en aangename samenwerking bij de oefenzittingen; v Christine Detroije en Anita Raets voor de vriendelijke en effici¨ente admini- stratieve ondersteuning; Martijn Derix en Toni Verbeiren voor hun compu- teradvies en hun geduld met mijn onwetendheid inzake computers. Graag wil ik ook Jeanne De Jaegher bedanken voor een toffe en geslaagde samenwerking; Kostas Skenderis voor de aanvullende wetenschappelijke be- geleiding; Frederik Roose voor de interessante discussies en aansporingen tot verruiming van mijn wetenschappelijke horizon; Yves Demasure voor de gea- nimeerde discussies over de dynamica van monopolen; Jos Gheerardyn voor de vragen die me op mijn eigen onbegrip wezen. Mijn dank gaat ook uit naar alle andere collega’s op het zesde: Mieke De Cock, voor het uitwisselen van ervaringen en frustraties tijdens het voltooien van het doctoraat; Michel Ver- schuere, Sorin Cucu, Pascal Spincemaille, Bart Haegeman, Joris Lauwers, Tom Michoel en Christian Maes, voor de vele conversaties, al of niet over de fysica, die gepaard gingen met de culinaire geneugten van de Alma; en alle andere collega’s voor de fijne interacties en geanimeerde discussies in het ‘koffiekot’. Graag zou ik ook alle leden van de sterrenkundige vereniging Wega bedan- ken, voor het stimuleren van mijn prille interesse voor de wetenschap, en om mij op jonge leeftijd reeds in contact te hebben gebracht met zulke exotische begrippen als zwarte gaten, snaren en extra dimensies. Het leven bestaat uiteraard niet alleen uit fysica; ‘de boog kan niet altijd gespannen staan’ zoals de volksmond zegt, en in dit verband wil ik ook mijn vrienden bedanken, in het bijzonder Dominique Coene en de andere trouwe Libertad-adepten, Henk Schets en Bieke Nijs, Leen Sammels, Katrien Kolen- berg, Leen Phalet: bedankt voor de caf´eavonden, kwissen, etentjes, feestjes, goede tot barslechte films en niet te vergeten een hoeveelheid e-mails die menige verantwoordelijke voor de harde schijfruimte tot wanhoop heeft ge- dreven. En, last but not least, wil ik mijn ouders bedanken, die me op ontelbare manieren hebben gesteund, zich de chronische kribbigheid die gepaard gaat met het voltooien van een doctoraat grootmoedig hebben laten welgevallen en zonder wier rustgevende aanwezigheid ondergetekende ongetwijfeld tot een neurotisch, ondervoed wrak was verworden: bedankt, Joris en Helga. vi Contents 1 Introduction 1 1.1 Stringtheory ................................ 1 1.2 Whystringfieldtheory? ......................... 5 1.3 Unstableobjectsinstringtheory . 9 1.4 Outlineofthethesis............................ 11 2 String perturbation theory 15 2.1 Polyakovaction............................... 15 2.2 Stringperturbationtheory . 17 2.2.1 Sumoverhistories .. ... ... ... ... ... ... ... 17 2.2.2 Gauge-fixing ............................ 20 2.3 Conformalfieldtheory . 23 2.3.1 OperatorProductExpansions . 23 2.3.2 Ward identities and conformal transformations . 25 2.3.3 Modeexpansionsandvertexoperators . 28 2.3.4 CFTontheupperhalfplane . 32 2.3.5 Innerproducts ........................... 32 2.3.6 GhostCFT.............................. 34 2.4 Physicalstates ............................... 37 2.4.1 BRSTformalism .......................... 38 vii Contents 2.4.2 Physicalvertexoperators . 39 2.5 Superstrings................................. 41 2.5.1 Gauge-fixedaction. 42 2.5.2 FermionicCFT ........................... 42 2.5.3 Superconformalsymmetry . 44 2.5.4 Superghosts............................. 45 2.5.5 Physicalstates ........................... 50 2.5.6 Otherconsistentstringtheories . 53 2.6 Summary................................... 54 3 Open string field theory 55 3.1 Bosonicopenstrings ........................... 56 3.1.1 The classical string field in various representations .... 56 3.1.2 Formalalgebraicstructure. 61 3.1.3 The ⋆ productandmidpointinteractions . 63 3.1.4 String field theory in the vertex operator representation . 67 3.1.5 Realitycondition. 76 3.1.6 Anexample: theactionforthelowestmodes . 77 3.1.7 Quantisationandmodularinvariance . 79 3.2 Opensuperstrings............................. 80 3.2.1 Witten’saction ........................... 82 3.2.2 Modifiedcubicactions . 84 3.2.3 Berkovits’action. 86 3.3 Open problems in present-daystring field theory . .. 88 4 Tachyoncondensationinstringtheory 91 4.1 Tachyonsandfieldtheoryinstabilities . .. 91 4.2 D-branesinstringtheory. 95 4.3 Dualitiesandstringsatstrongcoupling . .. 98 Contents 4.4 Non-BPS states in string theory and Sen’s conjecture . .... 101 4.4.1 Brane-antibranesystems . 101 4.4.2 Non-BPSD-branes . 102 4.4.3 Sen’sconjecture . 104 4.4.4 Tachyon condensation andstring fieldtheory . 104 5 Tachyoncondensationinbosonicstringfieldtheory 107 5.1 ThetachyonpotentialonaD25brane . 107 5.2 Universality ................................. 110 5.3 Theleveltruncationmethod . 111 5.4 Gauge-fixingandtwistinvariance . 111 5.5 Level4stringfield............................. 113 5.6 Level (4, 8) tachyonpotential . 113 5.7 Physicsaroundthestablevacuum . 118 5.8 Omissionsandfurtherdevelopments . 121 6 Toymodelfortachyoncondensation 123 6.1 Themodel.................................. 124 6.2 Equationforthecondensate. 125 6.3 Exact solution for µ 0 ......................... 126 = 6.4 The level truncation method and convergence issues . .. 130 7 Tachyon condensation in superstring theory 133 7.1 The tachyon potential in Witten’s superstring field theory .... 134 7.1.1 Non-BPSD-branesinWitten’stheory . 134 7.1.2 Thefieldsuptolevel2 . 138 7.1.3 Thetachyonpotential . 140 7.2 The tachyon potential in modified cubic theories . .. 142 7.2.1 Non-BPS D-branesin modifiedcubic theories . 142 Contents 7.2.2 Thetachyonpotential . 143 7.3 Tachyon condensationinBerkovits’theory . 150 7.3.1 Non-BPSD-branesinBerkovits’theory . 150 7.3.2 Thefieldsuptolevel2 . 152 7.3.3 Thetachyonpotential . 153 7.4 Conclusions................................. 154 A Conformaltransformationsofthefields 157 A.1 Witten’saction ............................... 157 A.2 Modifiedcubicstringfieldtheory . 158 A.3 Berkovits’theory.............................. 160 B Samenvatting 163 B.1 Situeringvanhetonderzoek. 163 B.1.1 Snaartheorie ............................ 163 B.1.2 Waaromsnaarveldentheorie? . 168 B.1.3 Onstabieleobjecteninsnaartheorie . 172 B.2 Overzichtvandethesis. 174 B.3 Samenvattingvanderesultaten. 179 Chapter 1 Introduction 1.1 String theory Unification One of the most important concepts in the development of theoretical physics is the search for unification. The unified description of diverse physical phe- nomena
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