A Short Introduction to String Theory

Preface
Acknowledgements
Introduction
Part I From Particles to Strings
1 Classical Relativistic Point Particles
1.1 Minkowski Space
1.2 Particles
1.3 A Non-covariant Action Principle for Relativistic Particles
1.4 Canonical Momenta and Hamiltonian
1.5 Length, Proper Time, and Reparametrisations
1.6 A Covariant Action for Massive Relativistic Particles
1.7 Particle Interactions
1.8 Canonical Momenta and Hamiltonian for the Covariant Action
1.9 A Covariant Action for Massless and Massive Particles
1.10 Literature
2 Classical Relativistic Strings
2.1 The Nambu–Goto Action
2.1.1 Action, Equations of Motion, and Bounday Conditions
2.1.2 D-branes
2.1.3 Constraints
2.2 The Polyakov Action
2.2.1 Action, Symmetries, Equations of Motion
2.2.2 Interpretation as a Two-Dimensional Field Theory
2.2.3 The Conformal Gauge
2.2.4 Light-Cone Coordinates
2.2.5 From Symmetries to Conservation Laws
2.2.6 Explicit Solutions – Periodic Boundary Conditions
2.2.7 Explicit Solutions – Neumann Boundary Conditions
2.2.8 Explicit Solutions – Dirichlet Boundary Conditions
2.2.9 Non-oriented Strings
2.2.10 Literature
3 Quantised Relativistic Particles and Strings
3.1 Quantised Relativistic Particles
3.2 Field Quantisation and Quantum Field Theory
3.3 Quantised Relativistic Strings
3.4 Literature on Quantum Field Theory
Part II The World-Sheet Perspective
4 The Free Massless Scalar Field on the Complex Plane
4.1 The Cylinder and the Plane
4.2 Infinitesimal and Finite Conformal Transformations
4.3 From Commutators to Operator Product Expansions
4.4 From Operators to States
5 Two-Dimensional Conformal Field Theories
5.1 Some Remarks on Conformal Field Theories in General Dimension
5.2 Conformal Primaries
5.2.1 Definition of Conformal Primaries, Holomorphic, and Chiral Fields
5.2.2 Hermitian Conjugation
5.2.3 Operator Products and Commutators
5.3 Energy-Momentum Tensor and Virasoro Algebra
5.4 The State – Operator Correspondence
5.4.1 The SL(2, C) Vacuum – Primary States
5.4.2 Highest Weight States
5.4.3 Descendant Fields
5.5 General Aspects of Two-Dimensional Conformal Field Theories
5.5.1 General Discussion of OPEs
5.5.2 Unitary Conformal Field Theories
5.5.3 Null Vectors
5.5.4 Results on Classification
5.6 Literature
6 Partition Functions I
6.1 Partition Functions for Particles and Strings
6.2 The Chiral Partition Function of a Free Boson
Part III The Space-Time Perspective
7 Covariant Quantisation I
7.1 Outline of Covariant Quantisation
7.2 The Fock Space
7.3 Implementation of the Constraints
7.4 Mass Eigenstates
7.5 Physical States of the Open String
7.6 The Photon
7.7 The Tachyon
7.8 Literature
8 Intermezzo – Representations of the Poincaré Group
8.1 Review of Representations of the Poincaré Group
8.2 Group Theoretical Interpretation of the Photon State
8.3 Virasoro Constraints, Poincaré Representations, and Effective Field Theory
8.4 Literature
9 Covariant Quantisation II
9.1 The Graviton
9.2 The Kalb–Ramond Field (B-Field)
9.2.1 Physical States
9.2.2 Dualisation of Antisymmetric Tensor Fields
9.3 Vertex Operators
9.4 The Dilaton
9.5 The No-Ghost Theorem
9.6 Further Remarks and Literature
10 Light-Cone Quantisation
10.1 Light-Cone Gauge and Light-Cone Quantisation for Particles
10.2 The Light-Cone Gauge for Open Strings
10.3 Light-Cone Quantisation for Open Strings
10.4 Ground State Energy via ζ-Function Method
10.5 Open String Spectrum in the Light-Cone Gauge
10.6 Lorentz Covariance in the Light-Cone Gauge
10.7 Literature
11 Partition Functions II
11.1 Partition Functions for Massive Particles
11.2 Partition Functions for Open Strings
11.3 Partition Functions for Closed Strings
11.4 Further Remarks and Literature
Part IV Outlook
12 Interactions
12.1 Amplitudes
12.1.1 General Discussion of Amplitudes
12.1.2 The Four Scalar Amplitude
12.1.3 The Four Graviton Amplitude
12.1.4 Effective Actions from Amplitudes
12.2 Curved Backgrounds
12.2.1 String Action for Curved Backgrounds and Background Fields
12.2.2 Conformal Invariance and the Equations of Motion
12.2.3 Consistent Backgrounds and Background Independence
12.2.4 Marginal Deformations
12.3 Further Remarks and Literature
13 Dimensional Reduction and T-Duality
13.1 Overview
13.2 Closed Strings on S1
13.2.1 Generic Massless States and Their Effective Field Theory
13.2.2 U(1) Gauge Symmetry from the World-Sheet Point of View
13.2.3 Enhanced Symmetries and Extra Massless States
13.2.4 SU(2) Gauge Symmetry from the World-Sheet Point of View
13.2.5 U(1) Gauge Symmetry, Higgs Effect, and Massive Vector Fields
13.2.6 SU(2)L × SU(2)R Gauge Symmetry from the Space-Time Point of View
13.2.7 The Higgs Effect from the World-Sheet Point of View
13.2.8 Partition Functions for Winding States
13.3 Closed Strings on S1 /Z2 and Orbifolds
13.4 T-Duality for Closed Strings
13.5 T-duality and Target Space Gauge Symmetries
13.6 T-duality for Open Strings
13.7 Toroidal Compactification of Closed Strings
13.7.1 The Narain Lattice
13.7.2 T-duality for Toroidal Compactifications
13.7.3 Symmetry Enhancement
13.7.4 Effective Actions for Toroidal Compactifications
13.7.5 Compactification on a Two-Torus
13.7.6 The Buscher Rules
13.8 The Vacuum Selection Problem
13.8.1 Beyond Toroidal Compactifications
13.8.2 From Moduli Spaces to the Landscape
13.8.3 The Landscape and the Swampland
13.9 Literature
14 Fermions and Supersymmetry
14.1 The Supersymmetric Harmonic Oscillator
14.1.1 The Simple Supersymmetric Harmonic Oscillator
14.1.2 Supersymmetrisation of the Light-Cone Hamiltonian
14.2 General Discussion of Supersymmetry
14.3 Supersymmetric Field Theories in Two Dimensions
14.4 The RNS String
14.5 Type-II Superstrings
14.6 Type-I Superstrings
14.7 Heterotic Strings
14.8 Looking for the Big Picture
14.9 Final Remarks and Literature
Appendix A Notation and Conventions
Appendix B Units, Constants, and Scales
Appendix C Fourier Series and Fourier Integrals
Appendix D Modular Forms and Special Functions
Appendix E Young Tableaux
Appendix F Gaussian Integrals and Integral Exponential Function
Appendix G Lie Algebras, Lie Groups, and Symmetric Spaces