A Short Introduction to String Theory

283 Pages·2022·3.3 MB·English

by  Thomas Mohaupt

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