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1 Introduction 1

1.1 Prehistory 1

1.2 The Case for String Theory 3

1.3 A Stringy Historical Perspective 6

1.4 Conventions 8

Bibliography 9

2 Classical String Theory 10

2.1 The Point Particle 10

2.2 Relativistic Strings 14

2.3 Oscillator Expansions 20

2.3.1 Closed strings 20

2.3.2 Open strings 22

2.3.3 The Virasoro constraints 24

Bibliography 26

Exercises 26

3 Quantization of Bosonic Strings 28

3.1 Covariant Canonic~Quantization 28

3.2 Light-cone Quantization 31

3.3 Spectrum ofthe Bosonic String 32

3.4 Unoriented Strings 33

3.4.1 Open strings and Chan-Paton factors 34

3.5 Path Integral Quantization 37

3.6 Topologically Nontrivial World-sheets 39

3.7 BRST Primer 40

3.8 BRST in String Theory and the Physical Spectrum 42

Bibliography 46

Exercises 46

4 Conformal Field Theory 49

4.1 Conformal Transformations 49

4.1.1 The case of two dimensions 51

4.2 Conformally Invariant Field Theory 52

4.3 Radial Quantization 54

4.4 Mode Expansions 57

4.5 The Virasoro Algebra and the Central Charge 58

4.6 The Hilbert Space 59

4.7 The Free Boson 60

4.8 The Free Fermion 63

4.9 The Conformal Anomaly 64

4.10 Representations ofthe Conformal Algebra 66

4.11 Affine Current Algebras 69

4.12 Free Fermions and O(N)Affine Symmetry 71

4.13 Superconformal Symmetry 77

4.13.1 N=(1,0)2 superconformal symmetry 77

4.13.2 N=(2,0)2 superconformalsymmetry 79

4.13.3 N=(4,0)2 superconformal symmetry 81

4.14 Scalars with Background Charge 82

4.15 The CFT of Ghosts 84

4.16 CFT on the Disk 86

4.16.1 Free massless bosons on the disk 86

4.16.2 Free massless fermions on the disk 88

4.16.3 The projective plane 90

4.17 CFT on the Torus 90

4.18 Compact Scalars 93

4.18.1 Modular invariance 97

4.18.2 Decompactification 97

4.18.3 The torus propagator 97

4.18.4 Marginal deformations 98

4.18.5 Multiple compact scalars 98

4.18.6 Enhanced symmetry and the string Brout-Englert-Higgs effect 100

4.18.7 T-duality 101

4.19 Free Fermions on the Torus 103

4.20 Bosonization 105

4.20.1"Bosonization"of bosonic ghost system 106

4.21 Orbifolds 107

4.22 CFT on Other Surfaces of Euler Number Zero 112

4.23 CFT on Higher-genus Riemann Surfaces 116

Bibliography 117

Exercises 118

5 Scattering Amplitudes and Vertex Operators 126

5.1 Physical Vertex Operators 128

5.2 Calculation of Tree-level Tachyon Amplitudes 130

5.2.1 The closed string 130

5.2.2 The open string 131

5.3 The One-loop Vacuum Amplitudes 133

5.3.1 The torus 134

5.3.2 The cylinder 136

5.3.3 The Klein bottle 138

5.3.4 The M？bius strip 138

5.3.5 Tadpole cancellation 139

5.3.6 UV structure and UV-IR correspondence 140

Bibliography 141

Exercises 142

6 Strings in Background Fields 144

6.1 The Nonlinearσ-model Approach 144

6.2 The Quest for Conformal Invariance 147

6.3 Linear Dilaton and Strings in D＜26 Dimensions 149

6.4 T-duality in Nontrivial Backgrounds 151

Bibliography 151

Exercises 152

7 Superstrings and Supersymmetry 155

7.1 N=(1,1)2 World-sheet Superconformal Symmetry 155

7.2 Closed(Type-Ⅱ)Superstrings 157

7.2.1 Massless R-R states 159

7.3 Type-Ⅰ Superstrings 162

7.4 Heterotic Superstrings 165

7.5 Superstring Vertex Operators 168

7.6 One-loop Superstring Vacuum Amplitudes 170

7.6.1 The type-IIA/B superstring 170

7.6.2 The heterotic superstring 171

7.6.3 The type-Ⅰsuperstring 171

7.7 Closed Superstrings and T-duality 174

7.7.1 The type-Ⅱ string theories 174

7.7.2 The heterotic string 175

7.8 Supersymmetric Effective Actions 175

7.9 Anomalies 176

Bibliography 182

Exercises 183

8 D-branes 187

8.1 Antisymmetric Tensors and p-branes 187

8.2 Open Strings and T-duality 188

8.3 D-branes 191

8.4 D-branes and R-R Charges 193

8.4.1 D-instantons 196

8.5 D-brane Efiective Actions 197

8.5.1 The Dirac-Born-Infeld action 197

8.5.2 Anomaly-related terms 199

8.6 Multiple Branes and Nonabelian Symmetry 200

8.7 T-duality and Orientifolds 201

8.8 D-branes as Supergravity Solitons 205

8.8.1 The supergravity solutions 205

8.8.2 Horizons and singularities 207

8.8.3 The extremal branes and their near-horizongeometry 208

8.9 NS5-branes 211

Bibliography 213

Exercises 213

9 Compactifications and Supersymmetry Breaking 219

9.1 Narain Compactifications 219

9.2 World-sheet versus Space-time Supersymmetry 223

9.2.1 N=14 space-time supersymmetry 225

9.2.2 N=24 space-time supersymmetry 226

9.3 Orbifold Reduction of Supersymmetry 228

9.4 A Heterotic Orbifold with N=24 Supersymmetry 231

9.5 Spontaneous Supersymmetry Breaking 235

9.6 A Heterotic N=14 Orbifold and Chirality in Four Dimensions 237

9.7 Calabi-Yau Manifolds 239

9.7.1 Holonomy 241

9.7.2 Consequences of SU(3)holonomy 242

9.7.3 The CY moduli space 243

9.8 N=14 Heterotic Compactifications 245

9.8.1 The low-energy N=14 heterotic spectrum 246

9.9 K3 Compactification of the Type-Ⅱ String 247

9.10 N=26 Orbifolds of the Type-Ⅱ String 250

9.11 CY Compactifications of Type-Ⅱ Strings 252

9.12 Mirror Symmetry 253

9.13 Absence of Continuous Global Symmetries 255

9.14 Orientifolds 256

9.14.1 K3 orientifolds 257

9.14.2 The Klein bottle amplitude 258

9.14.3 D-branes on T4/Z2 260

9.14.4 The cylinder amplitude 263

9.14.5 The M？bius strip amplitude 265

9.14.6 Tadpole cancellation 266

9.14.7 The open string spectrum 267

9.15 D-branes at Orbifold Singularities 268

9.16 Magnetized Compactifications and Intersecting Branes 271

9.16.1 Open strings in an internal magneticfield 272

9.16.2 Intersecting branes 277

9.16.3 Intersecting D6-branes 278

9.17 Where is the Standard Model? 280

9.17.1 The heterotic string 280

9.17.2 Type-Ⅱ string theory 282

9.17.3 The type-Ⅰstring 283

9.18 Unification 284

Bibliography 286

Exercises 287

10 Loop Corrections to String Effective Couplings 294

10.1 Calculation of Heterotic Gauge Thresholds 296

10.2 On-shell Infrared Regularization 301

10.2.1 Evaluation of the threshold 303

10.3 Heterotic Gravitational Thresholds 304

10.4 One-loop Fayet-Iliopoulos Terms 305

10.5 N=1,24 Examples of Threshold Corrections 309

10.6 N=24 Universality of Thresholds 312

10.7 Unification Revisited 315

Bibliography 317

Exercises 317

11 Duality Connections and Nonperturbative Effects 320

11.1 Perturbative Connections 322

11.2 BPS States and BPS Bounds 323

11.3 Nonrenormalization Theorems and BPS-saturated Couplings 325

11.4 Type-ⅡA versus M-theory 328

11.5 Self-duality of the Type-ⅡB String 331

11.6 U-duality of Type-Ⅱ String Theory 334

11.6.1 U-duality and bound states 336

11.7 Heterotic/Type Ⅰ Duality in Ten Dimensions 336

11.7.1 The type-Ⅰ D1-string 339

11.7.2 The type-Ⅰ D5-brane 341

11.7.3 Further consistency checks 343

11.8 M-theory and the E8×E8 Heterotic String 344

11.8.1 Unification at strong heterotic coupling 347

11.9 Heterotic/Type Ⅱ Duality in Six Dimensions 348

11.9.1 Gauge synmetry enhancementand singular K3 surfaces 352

11.9.2 Heterotic/type Ⅱ duality in four dimensions 355

11.10 Conifold Singularities and Conifold Transitions 356

Bibliography 362

Exercises 363

12 Black Holes and Entropy in String Theory 369

12.1 A Brief History 369

12.2 The Strategy 370

12.3 Black-hole Thermodynamics 371

12.3.1 The Euclidean continuation 372

12.3.2 Hawking evaporation and greybody factors 374

12.4 The Information Problem and the Holographic Hypothesis 375

12.5 Fire-dimensional Extremal Charged Black Holes 377

12.6 Five-dimensional Nonextremal RN Black Holes 379

12.7 The Near-horizon Region 381

12.8 Semiclassical Derivation of the Hawking Rate 383

12.9 The Microscopic Realization 386

12.9.1 Theworld-volume theory of the bound state 387

12.9.2 The low-energy SCFT of the Dr-D5 bound state 389

12.9.3 Microscopic calculation of the entropy 391

12.9.4 Microscopic derivation of Hawking evaporation rates 394

12.10 Epilogue 396

Bibliography 398

Exercises 399

13 The Bulk/Boundary Correspondence 403

13.1 Large-N Gauge Theories and String Theory 405

13.2 The Decoupling Principle 408

13.3 The Near-horizon Limit 409

13.4 Elements of the Correspondence 410

13.5 Bulk Fields and Boundary Operators 413

13.6 Holography 416

13.7 Testing the AdS5/CFT4 Correspondence 417

13.7.1 The chiral spectrum of N=4 gaugetheory 418

13.7.2 Matching to the string theory spectrum 420

13.7.3 N=8 five-dimensional gauged supergravity 422

13.7.4 Protected correlation functions and anomalies 422

13.8 Correlation Functions 424

13.8.1 Two-pointfunctions 425

13.8.2 Three-pointfunctions 427

13.8.3 The gravitational action and the conformal anomaly 428

13.9 Wilson Loops 433

13.10 AdS5/CFT4 Correspondence at Finite Temperature 436

13.10.1 N=4 super yang-Mills theory atfinite temperature 436

13.10.2 The near-horizon limit of black D3-branes 438

13.10.3 Finite-volume and large-Nphase transitions 440

13.10.4 Thermal holographic physics 443

13.10.5 Spatial Wilson loops in(a version of)QCD3 444

13.10.6 The glueball mass spectrum 446

13.11 AdS3/CFT2 Correspondence 447

13.11.1 The greybodyfuctors revisited 450

13.12 The Holographic Renormalization Group 450

13.12.1 Perturbations of the CFT4 451

13.12.2 Domain walls and flow equations 452

13.12.3 A RG flow preserving N=1 supersymmetry 454

13.13 The Randall-Sundrum Geometry 456

13.13.1 An alternative to compactification 459

Bibliography 462

Exercises 463

14 String Theory and Matrix Models 470

14.1 M(atrix) Theory 471

14.1.1 Membrane quantization 471

14.1.2 Type-ⅡA Do-branes and DLCQ 473

14.1.3 Gravitons and branes in M(atrix)theory 476

14.1.4 The two-graviton interaction from M(atrix) theory 477

14.2 Matrix Models and D=1 Bosonic String Theory 479

14.2.1 The continuum limit 481

14.2.2 Solving the matrix model 482

14.2.3 The double-scaling limit 485

14.2.4 The free-fermion picture 487

14.3 Matrix Description of D=2 String Theory 488

14.3.1 Matrix quantum mechanics and free fermions on the line 490

14.3.2 The continuum limit 492

14.3.3 The double-scaling limit 494

14.3.4 D-particles,tachyons,and holography 496

Bibliography 498

Exercises 498

Appendix A Two-dimensional Complex Geometry 503

Appendix B Differential Forms 505

Appendix C Theta and Other Elliptic Functions 507

C.1 v and Related Functions 507

C.2 The Weierstrass Function 510

C.3 Modular Forms 510

C.4 Poisson Resummation 512

Appendix D Toroidal Lattice Sums 513

Appendix E Toroidal Kaluza-Klein Reduction 516

Appendix F The Reissner-Nordstr？m Black Hole 519

Appendix G Electric-magnetic Duality in D=4 522

Appendix H Supersymmetric Actions in Ten and Eleven Dimensions 525

H.1 The N=111 Supergravity 526

H.2 Type-ⅡA Supergravity 527

H.3 Type-ⅡB Supergravity 528

H.4 Type-Ⅱ Supergravities：The Democratic Formulation 529

H.5 N=110 Supersymmetry 530

Appendix Ⅰ N=1,2,Four-dimensional Supergravity Coupled to Matter 533

I.1 N=14 Supergravity 533

I.2N=24 Supergravity 535

Appendix J BPS Multiplets in Four Dimensions 537

Appendix K The Geometry of Anti-de Sitter Space 541

K.1 The Minkowski Signature AdS 541

K.2 Euclidean AdS 544

K.3 The Conformal Structure of Flat Space 546

K.4 Fields in AdS 548

K.4.1 The wave equation in Poincaré coordinates 549

K.4.2 The bulk-boundary propagator 550

K.4.3 The bulk-to-bulk propagator 551

Bibliography 553

Index 575