Introduction to String Theory A.N. Schellekens Based on lectures given at the Radboud Universiteit, Nijmegen Last update 6 July 2016 [Word cloud by www.worldle.net] Contents 1 Current Problems in Particle Physics7 1.1 Problems of Quantum Gravity.........................9 1.2 String Diagrams................................. 11 2 Bosonic String Action 15 2.1 The Relativistic Point Particle......................... 15 2.2 The Nambu-Goto action............................ 16 2.3 The Free Boson Action............................. 16 2.4 World sheet versus Space-time......................... 18 2.5 Symmetries................................... 19 2.6 Conformal Gauge................................ 20 2.7 The Equations of Motion............................ 21 2.8 Conformal Invariance.............................. 22 3 String Spectra 24 3.1 Mode Expansion................................ 24 3.1.1 Closed Strings.............................. 24 3.1.2 Open String Boundary Conditions................... 25 3.1.3 Open String Mode Expansion..................... 26 3.1.4 Open versus Closed........................... 26 3.2 Quantization.................................. 26 3.3 Negative Norm States............................. 27 3.4 Constraints................................... 28 3.5 Mode Expansion of the Constraints...................... 28 3.6 The Virasoro Constraints............................ 29 3.7 Operator Ordering............................... 30 3.8 Commutators of Constraints.......................... 31 3.9 Computation of the Central Charge...................... 31 3.10 The Virasoro Algebra.............................. 32 3.11 Imposing the Virasoro Constraints...................... 33 3.12 The Mass Shell Condition........................... 34 3.12.1 Closed Strings.............................. 34 3.12.2 Open Strings.............................. 34 3.12.3 Tachyons?................................ 34 3.13 Unphysical State Decoupling.......................... 35 3.14 Light-Cone Gauge................................ 36 3.14.1 Constraints in Light-Cone Coordinates................ 37 3.14.2 Zero-Mode Constraints: Closed Strings................ 38 3.14.3 Zero-Mode Constraints: Open Strings................. 39 3.15 The Spectrum.................................. 39 3.15.1 Open Strings.............................. 39 2 3.15.2 The Little Group............................ 40 3.15.3 Tensor Representations of Orthogonal Groups............ 41 3.15.4 Open Strings (continued)........................ 42 3.15.5 Closed Strings.............................. 42 3.16 The Critical Dimension............................. 43 3.17 The Lorentz Algebra∗ .............................. 44 3.18 Unoriented Strings∗ ............................... 44 3.19 Chan-Paton labels∗ ............................... 45 3.20 Discussion.................................... 46 4 Compactification 49 4.1 Space-time Compactification for Particles................... 49 4.2 Space-time Compactification for Strings.................... 50 4.3 The Spectrum.................................. 50 4.4 T-duality.................................... 51 4.5 The Self-Dual Point∗ .............................. 52 4.6 Field Theory Interpretation∗ .......................... 53 4.7 Compactification of More Dimensions..................... 54 4.8 Narain Lattices................................. 56 4.9 Background Fields............................... 57 4.9.1 The Metric............................... 57 4.9.2 The Anti-Symmetric Tensor...................... 57 4.10 Moduli∗ ..................................... 59 4.11 Vacuum Selection∗ ............................... 60 4.12 Orbifolds∗ .................................... 62 5 Fermions on the world sheet 63 5.1 Generalizations of the Bosonic String..................... 63 5.2 The R^oleof the Conformal Anomaly∗ ..................... 64 5.3 Internal Free Fermions............................. 65 5.4 Quantization.................................. 66 5.5 The Energy-Momentum Tensor........................ 68 5.6 Noether Currents................................ 69 5.7 The Noether Current of Translation Invariance................ 69 5.8 The Mass Spectrum............................... 71 5.9 The Neveu-Schwarz Ground State....................... 72 5.10 The Ramond Ground State.......................... 72 5.11 Excited States.................................. 73 5.11.1 Neveu-Schwarz Sector......................... 73 5.11.2 Ramond Sector............................. 74 5.11.3 The Lorentz Algebra.......................... 74 5.12 Fermion-Boson Equivalence∗ .......................... 75 5.13 Fermionic Strings................................ 75 3 5.14 The Critical Dimension............................. 76 5.15 The Lorentz Algebra∗ .............................. 77 5.16 The Ramond Ground State.......................... 77 5.17 World Sheet Supersymmetry∗ ......................... 78 5.18 World Sheet Supergravity∗ ........................... 78 5.19 The Constraints∗ ................................ 79 6 One Loop Diagrams 80 6.1 Neveu-Schwarz or Ramond?.......................... 80 6.2 The Torus.................................... 80 6.3 The Polyakov Path Integral∗ .......................... 81 6.3.1 The Conformal Anomaly........................ 83 6.4 Sum over Topologies.............................. 83 6.4.1 Riemann Surfaces∗ ........................... 83 6.4.2 Topological Numbers.......................... 84 6.5 The R^oleof the Dilaton∗ ............................ 85 6.5.1 Bosonic String in a Dilaton Background............... 85 6.5.2 The String Coupling Constant..................... 85 6.6 Moduli...................................... 86 6.6.1 Moduli of Riemann Surfaces∗ ..................... 86 6.6.2 Moduli of the Torus.......................... 87 6.7 Complex Coordinates.............................. 88 6.8 Modular Transformations............................ 88 6.9 Integration over Moduli............................ 90 6.10 Computing the Path Integral......................... 91 6.11 The Bosonic String Partition Function.................... 92 6.11.1 Partition Functions and Physical State Counting........... 92 6.11.2 Assembling the Bosonic String Partition Function.......... 94 6.11.3 The Contribution of Momenta..................... 95 6.12 Modular Invariance of the Bosonic String................... 95 6.13 Strings versus Particles∗ ............................ 96 6.14 Partition Function for Compactified Strings................. 99 7 Superstrings 102 7.1 Free Fermion Partition Functions....................... 102 7.1.1 Neveu-Schwarz states.......................... 102 7.1.2 Ramond States............................. 103 7.2 Boundary Conditions.............................. 103 7.3 Modular Transformations............................ 106 7.4 The Modular Invariant Partition Function.................. 107 7.5 Multi-loop Modular Invariance∗ ........................ 108 7.6 Superstrings................................... 109 7.7 Supersymmetry and Supergravity∗ ...................... 110 4 7.8 Supersymmetry for the Massive States.................... 110 7.9 Ramond-Ramond Particles∗ .......................... 111 7.10 Other Modular Invariant String Theories................... 111 7.11 Internal Fermions................................ 112 7.11.1 Modular Invariant Partition Functions for Internal Free Fermions. 112 7.11.2 Factorized Partition Functions..................... 113 7.11.3 Symmetries of Free Fermions...................... 114 7.11.4 The 18- and 10-Dimensional Theories................. 114 7.11.5 Relation with Even Self-dual Lorentzian Lattices.......... 115 8 Heterotic Strings 116 8.1 The Massless Spectrum............................. 117 8.2 Comparison with other Closed String Theories................ 117 8.3 Supersymmetry∗ ................................ 117 8.4 Gauge symmetry∗ ................................ 118 8.5 Compactification∗ ................................ 118 8.5.1 Breaking the Lorentz Group∗ ..................... 119 8.5.2 Decomposition of Spinors∗ ....................... 119 8.5.3 Reduction of the Dirac Equation∗ ................... 121 8.6 Calabi-Yau Compactification∗ ......................... 122 8.7 The Standard Model in String Theory∗ .................... 123 8.8 Other Ten-dimensional Heterotic Strings................... 126 8.9 Four-dimensional Strings∗ ........................... 126 9 The String Theory Landscape and the Multiverse∗ 127 9.1 Moduli and Supersymmetry Breaking..................... 127 9.1.1 Space-time Supersymmetry...................... 128 9.1.2 Moduli Stabilization.......................... 128 9.2 A Proliferation of Vacua............................ 129 9.2.1 Vacuum Energy............................. 129 9.2.2 Anthropic Arguments.......................... 130 9.2.3 Eternal Inflation............................ 131 9.3 A Change of Perspective............................ 131 10 Dualities∗ 131 10.1 Open Strings.................................. 132 10.2 Finiteness.................................... 132 10.3 Dual Models................................... 134 10.4 Historical Remarks............................... 137 10.5 T-duality.................................... 138 10.6 Perturbative and Non-perturbative Physics.................. 138 10.7 D-branes..................................... 139 10.8 S-duality..................................... 141 5 Appendices 143 A Functional Methods 143 A.1 The Euclidean Action.............................. 146 B Path Integrals 146 C Self-dual Lattices 148 D Lie Algebra Dictionary 150