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Introduction to String Theory

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Introduction to String Theory Introduction to String Theory Angel M. Uranga Contents I Introductory Overview 1 1 Motivation 3 1.1 Standard Model and beyond . 3 1.1.1 Our Model of Elementary Particles and Interactions . 3 1.1.2 Theoretical questions raised by this description . 5 1.1.3 Some proposals for physics beyond the Standard Model 7 1.1.4 String theory as a theory beyond the Standard Model . 13 2 Overview of string theory in perturbation theory 15 2.1 Basic ideas . 15 2.1.1 What are strings? . 15 2.1.2 The worldsheet . 17 2.1.3 String interactions . 19 2.1.4 Critical dimension . 23 2.1.5 Overview of closed bosonic string theory . 24 2.1.6 String theory in curved spaces . 26 2.1.7 Compactification . 31 2.2 Superstrings and Heterotic string phenomenology . 34 2.2.1 Superstrings . 34 2.2.2 Heterotic string phenomenology . 39 2.2.3 The picture of our world as a heterotic string compact- ification . 40 2.2.4 Phenomenological features and comparison with other proposals beyond the standard model . 42 3 Overview of string theory beyond perturbation theory 45 3.1 The problem . 45 3.2 Non-perturbative states in string theory . 47 i ii CONTENTS 3.2.1 Non-perturbative states in field theory . 47 3.2.2 Non-perturbative p-brane states in string theory . 52 3.2.3 Duality in string theory . 55 3.3 D-branes . 62 3.3.1 What are D-branes . 62 3.3.2 Worldvolume theory . 64 3.3.3 D-branes in string theory . 66 3.3.4 D-branes as probes of spacetime . 69 3.3.5 D-branes and gauge field theories . 71 3.4 Our world as a brane-world model . 72 4 Quantization of the closed bosonic string 77 4.1 Worldsheet action . 77 4.1.1 The Nambu-Goto action . 78 4.1.2 The Polyakov action . 78 4.1.3 Symmetries of Polyakov action . 79 4.2 Light-cone quantization . 80 4.2.1 Light-cone gauge fixing . 80 4.2.2 Gauge-fixed Polyakov action, Hamiltonian . 83 4.2.3 Oscillator expansions . 85 4.2.4 Light spectrum . 88 4.2.5 Lessons . 89 4.2.6 Final comments . 90 5 Modular invariance 91 5.1 Generalities . 91 5.2 Worldsheet coordinatization in light-cone gauge . 92 5.3 The computation . 93 5.3.1 Structure of the amplitude in operator formalism . 93 5.3.2 The momentum piece . 95 5.3.3 The oscillator piece . 95 5.4 Modular invariance . 96 5.4.1 Modular group of T2 . 96 5.4.2 Modular invariance of the partition function . 99 5.4.3 UV behaviour of the string amplitude . 100 CONTENTS iii 6 Toroidal compactification of closed bosonic string theory 105 6.1 Motivation . 105 6.2 Toroidal compactification in field theory . 106 6.3 Toroidal compactification in string theory . 110 6.3.1 Quantization and spectrum . 111 6.3.2 α0 effects I: Enhanced gauge symmetries . 117 6.3.3 α0 effects II: T-duality . 120 6.3.4 Additional comments . 123 7 Type II Superstrings 125 7.1 Superstrings . 125 7.1.1 Fermions on the worldsheet . 125 7.1.2 Boundary conditions . 127 7.1.3 Spectrum of states for NS and R fermions . 128 7.1.4 Modular invariance . 133 7.1.5 Type II superstring partition function . 135 7.1.6 GSO projection . 135 7.1.7 Light spectrum . 136 7.2 Type 0 superstrings . 140 7.3 Bosonization∗ . 141 8 Heterotic superstrings 145 8.1 Heterotic superstrings in bosonic formulation . 145 8.1.1 Heteroticity . 145 8.1.2 Hamiltonian quantization . 146 8.1.3 Modular invariance and lattices . 148 8.1.4 Spectrum . 151 8.2 Heterotic strings in the fermionic formulation . 154 8.3 Spacetime Non-susy heterotic string theories . 158 8.4 A few words on anomalies . 159 8.4.1 What is an anomaly? . 159 8.4.2 Anomalies in string theory and Green-Schwarz mech- anism . 162 9 Open strings 165 9.1 Generalities . 165 9.2 Open bosonic string . 167 9.2.1 Light-cone gauge . 167 iv CONTENTS 9.2.2 Boundary conditions . 168 9.2.3 Hamiltonian . 169 9.2.4 Oscillator expansions . 170 9.2.5 Spectrum . 171 9.2.6 Open-closed duality . 172 9.3 Chan-Paton factors . 175 9.4 Open superstrings . 177 9.4.1 Hamiltonian quantization . 177 9.4.2 Spectrum for NS and R sectors . 179 9.4.3 GSO projection . 180 9.4.4 Open-closed duality . 180 9.4.5 RR tadpole cancellation condition . 181 10 Type I superstring 185 10.1 Unoriented closed strings . 185 10.1.1 Generalities . 185 10.1.2 Unoriented closed bosonic string . 187 10.1.3 Unoriented closed superstring theory IIB/Ω . 188 10.2 Unoriented open strings . 191 10.2.1 Action of Ω on open string sectors . 191 10.2.2 Spectrum . 192 10.3 Type I superstring . 193 10.3.1 Computation of RR tadpoles . 193 10.4 Final comments . 200 11 Toroidal compactification of superstrings 203 11.1 Motivation . 203 11.2 Type II superstrings . 203 11.2.1 Circle compactification . 203 11.2.2 T-duality for type II theories . 208 11.2.3 Compactification of several dimensions . 210 11.3 Heterotic superstrings . 215 11.3.1 Circle compactification without Wilson lines . 215 11.3.2 Compactification with Wilson lines . 218 11.3.3 Field theory description of Wilson lines . 218 11.3.4 String theory description . 221 11.4 Toroidal compactification of type I superstring . 226 11.4.1 Circle compactification without Wilson lines . 227 CONTENTS v 11.4.2 T-duality . 229 11.4.3 Toroidal compactification and T-duality in type I with Wilson lines . 234 11.5 Final comments . 238 12 Calabi-Yau compactification of superstrings. Heterotic string phenomenology 239 12.1 Motivation . 239 12.1.1 Supersymmetry and holonomy . 240 12.1.2 Calabi-Yau manifolds . 242 12.2 Type II string theories on Calabi-Yau spaces . 246 12.2.1 Supersymmetry . 246 12.2.2 KK reduction of p-forms . 247 12.2.3 Spectrum . 248 12.2.4 Mirror symmetry . 249 12.3 Compactification of heterotic strings on Calabi-Yau threefolds 250 12.3.1 General considerations . 250 12.3.2 Spectrum . 253 12.3.3 Phenomenological features of these models . 257 13 Orbifold compactification 261 13.1 Introduction . 261 13.1.1 Motivation . 261 13.1.2 The geometry of orbifolds . 262 13.1.3 Generalities of string theory on orbifolds . 265 6 13.2 Type II string theory on T =Z3 . 268 13.2.1 Geometric interpretation . 274 6 13.3 Heterotic string compactification on T =Z3 . 275 13.3.1 Gauge bundles for orbifolds . 275 13.3.2 Computation of the spectrum . 276 13.3.3 Final comments . 279 14 Non-perturbative states in string theory 281 14.1 Motivation . 281 14.2 p-branes in string theory . 281 14.2.1 p-brane solutions . 283 14.2.2 Dirac charge quantization condition . 287 14.2.3 BPS property . 288 vi CONTENTS 14.3 Duality for type II string theories . 289 14.3.1 Type IIB SL(2; Z) duality . 290 14.3.2 Toroidal compactification and U-duality . 291 14.4 Final comments . 294 .1 Some similar question in the simpler context of field theory . 295 .1.1 States in field theory . 295 .1.2 BPS bounds . ..
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