AShortIntroductiontoStringTheory Suitable for graduate students in physics and mathematics, this book presents a concise and pedagogical introduction to string theory. It focuses on explaining the key concepts of string theory, such as bosonic strings, D-branes, supersymmetry, and superstrings and on clarifying the relationship between particles, fields, and strings without assuming an advanced background in particle theory or quantum field theory, thus making it widely accessible to interested readers from a range of backgrounds. Important ideas underpinning current research, such as partition functions,compactification,gaugesymmetries,andT-dualityareanalysedbothfrom the world-sheet (conformal field theory) and the space-time (effective field theory) perspectives. Ideal for either self-study or a one semester graduate course, A Short Introduction to String Theory is an essential resource for students studying string theory,containingexamplesandhomeworkproblemstodevelopunderstanding,with fullyworkedsolutionsavailabletoinstructors. ThomasMohaupt obtained his PhD in Theoretical Physics from the University of Mu¨nster and is Reader in Theoretical Physics at the University of Liverpool. He has published extensively on string theory, specialising in black hole entropy and supergravity, and has taught various courses on related topics in the UK and Germany.HehasbeenavisitingscholaratStanfordUniversity,avisitingprofessorat theUniversityofNancy,andaSeniorFellowattheErwinSchro¨dingerInternational InstituteforMathematicsandPhysicsinVienna. A Short Introduction to String Theory THOMAS MOHAUPT UniversityofLiverpool UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108481380 DOI:10.1017/9781108611619 ©ThomasMohaupt2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. ISBN978-1-108-48138-0Hardback Additionalresourcesforthispublicationatwww.cambridge.org/stringtheory. CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Thisbookisdedicatedtomyparents,DrHelgaandManfredMohaupt. Contents Preface pagexiii Acknowledgements xv Introduction xvi PartI FromParticlestoStrings 1 1 ClassicalRelativisticPointParticles 3 1.1 MinkowskiSpace 3 1.2 Particles 5 1.3 ANon-covariantActionPrincipleforRelativisticParticles 6 1.4 CanonicalMomentaandHamiltonian 9 1.5 Length,ProperTime,andReparametrisations 9 1.6 ACovariantActionforMassiveRelativisticParticles 11 1.7 ParticleInteractions 12 1.8 CanonicalMomentaandHamiltonianfortheCovariantAction 14 1.9 ACovariantActionforMasslessandMassiveParticles 15 1.10 Literature 17 2 ClassicalRelativisticStrings 18 2.1 TheNambu–GotoAction 18 2.1.1 Action,EquationsofMotion,andBoundayConditions 18 2.1.2 D-branes 21 2.1.3 Constraints 23 2.2 ThePolyakovAction 24 2.2.1 Action,Symmetries,EquationsofMotion 24 2.2.2 InterpretationasaTwo-DimensionalFieldTheory 26 2.2.3 TheConformalGauge 27 2.2.4 Light-ConeCoordinates 28 2.2.5 FromSymmetriestoConservationLaws 29 2.2.6 ExplicitSolutions–PeriodicBoundaryConditions 32 2.2.7 ExplicitSolutions–NeumannBoundaryConditions 34 2.2.8 ExplicitSolutions–DirichletBoundaryConditions 35 2.2.9 Non-orientedStrings 36 2.2.10 Literature 37 3 QuantisedRelativisticParticlesandStrings 38 3.1 QuantisedRelativisticParticles 38 3.2 FieldQuantisationandQuantumFieldTheory 41 vii viii Contents 3.3 QuantisedRelativisticStrings 43 3.4 LiteratureonQuantumFieldTheory 46 PartII TheWorld-SheetPerspective 47 4 TheFreeMasslessScalarFieldontheComplexPlane 49 4.1 TheCylinderandthePlane 49 4.2 InfinitesimalandFiniteConformalTransformations 51 4.3 FromCommutatorstoOperatorProductExpansions 53 4.4 FromOperatorstoStates 59 5 Two-DimensionalConformalFieldTheories 63 5.1 SomeRemarksonConformalFieldTheoriesinGeneral Dimension 63 5.2 ConformalPrimaries 65 5.2.1 Definition of Conformal Primaries, Holomorphic, and ChiralFields 65 5.2.2 HermitianConjugation 67 5.2.3 OperatorProductsandCommutators 68 5.3 Energy-MomentumTensorandVirasoroAlgebra 69 5.4 TheState–OperatorCorrespondence 72 5.4.1 TheSL(2,C)Vacuum–PrimaryStates 72 5.4.2 HighestWeightStates 73 5.4.3 DescendantFields 74 5.5 GeneralAspectsofTwo-DimensionalConformalFieldTheories 75 5.5.1 GeneralDiscussionofOPEs 75 5.5.2 UnitaryConformalFieldTheories 75 5.5.3 NullVectors 76 5.5.4 ResultsonClassification 76 5.6 Literature 77 6 PartitionFunctionsI 78 6.1 PartitionFunctionsforParticlesandStrings 78 6.2 TheChiralPartitionFunctionofaFreeBoson 81 PartIII TheSpace-TimePerspective 85 7 CovariantQuantisationI 87 7.1 OutlineofCovariantQuantisation 87 7.2 TheFockSpace 87 7.3 ImplementationoftheConstraints 88 7.4 MassEigenstates 90 7.5 PhysicalStatesoftheOpenString 91 7.6 ThePhoton 92 7.7 TheTachyon 95 7.8 Literature 98 ix Contents 8 Intermezzo–RepresentationsofthePoincaréGroup 99 8.1 ReviewofRepresentationsofthePoincare´ Group 99 8.2 GroupTheoreticalInterpretationofthePhotonState 101 8.3 Virasoro Constraints, Poincare´ Representations, and Effective FieldTheory 102 8.4 Literature 103 9 CovariantQuantisationII 104 9.1 TheGraviton 104 9.2 TheKalb–RamondField(B-Field) 108 9.2.1 PhysicalStates 108 9.2.2 DualisationofAntisymmetricTensorFields 109 9.3 VertexOperators 111 9.4 TheDilaton 112 9.5 TheNo-GhostTheorem 114 9.6 FurtherRemarksandLiterature 116 10 Light-ConeQuantisation 117 10.1 Light-ConeGaugeandLight-ConeQuantisationforParticles 117 10.2 TheLight-ConeGaugeforOpenStrings 118 10.3 Light-ConeQuantisationforOpenStrings 119 10.4 GroundStateEnergyviaζ-FunctionMethod 120 10.5 OpenStringSpectrumintheLight-ConeGauge 121 10.6 LorentzCovarianceintheLight-ConeGauge 122 10.7 Literature 124 11 PartitionFunctionsII 125 11.1 PartitionFunctionsforMassiveParticles 125 11.2 PartitionFunctionsforOpenStrings 127 11.3 PartitionFunctionsforClosedStrings 131 11.4 FurtherRemarksandLiterature 132 PartIV Outlook 135 12 Interactions 137 12.1 Amplitudes 137 12.1.1 GeneralDiscussionofAmplitudes 137 12.1.2 TheFourScalarAmplitude 140 12.1.3 TheFourGravitonAmplitude 143 12.1.4 EffectiveActionsfromAmplitudes 144 12.2 CurvedBackgrounds 146 12.2.1 StringActionforCurvedBackgroundsand BackgroundFields 146 12.2.2 ConformalInvarianceandtheEquationsofMotion 148 12.2.3 ConsistentBackgroundsandBackgroundIndependence 149 12.2.4 MarginalDeformations 152 12.3 FurtherRemarksandLiterature 153