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1 Introduction1

1.1 Prehistory1

1.2 The Case for String Theory3

1.3 A Stringy Historical Perspective6

1.4 Conventions8

Bibliography9

2 Classical String Theory10

2.1 The Point Particle10

2.2 Relativistic Strings14

2.3 Oscillator Expansions20

2.3.1 Closed strings20

2.3.2 Open strings22

2.3.3 The Virasoro constraints24

Bibliography26

Exercises26

3 Quantization of Bosonic Strings28

3.1 Covariant Canonic~Quantization28

3.2 Light-cone Quantization31

3.3 Spectrum ofthe Bosonic String32

3.4 Unoriented Strings33

3.4.1 Open strings and Chan-Paton factors34

3.5 Path Integral Quantization37

3.6 Topologically Nontrivial World-sheets39

3.7 BRST Primer40

3.8 BRST in String Theory and the Physical Spectrum42

Bibliography46

Exercises46

4 Conformal Field Theory49

4.1 Conformal Transformations49

4.1.1 The case of two dimensions51

4.2 Conformally Invariant Field Theory52

4.3 Radial Quantization54

4.4 Mode Expansions57

4.5 The Virasoro Algebra and the Central Charge58

4.6 The Hilbert Space59

4.7 The Free Boson60

4.8 The Free Fermion63

4.9 The Conformal Anomaly64

4.10 Representations ofthe Conformal Algebra66

4.11 Affine Current Algebras69

4.12 Free Fermions and O(N)Affine Symmetry71

4.13 Superconformal Symmetry77

4.13.1 N=(1,0)2 superconformal symmetry77

4.13.2 N=(2,0)2 superconformalsymmetry79

4.13.3 N=(4,0)2 superconformal symmetry81

4.14 Scalars with Background Charge82

4.15 The CFT of Ghosts84

4.16 CFT on the Disk86

4.16.1 Free massless bosons on the disk86

4.16.2 Free massless fermions on the disk88

4.16.3 The projective plane90

4.17 CFT on the Torus90

4.18 Compact Scalars93

4.18.1 Modular invariance97

4.18.2 Decompactification97

4.18.3 The torus propagator97

4.18.4 Marginal deformations98

4.18.5 Multiple compact scalars98

4.18.6 Enhanced symmetry and the string Brout-Englert-Higgs effect100

4.18.7 T-duality101

4.19 Free Fermions on the Torus103

4.20 Bosonization105

4.20.1"Bosonization"of bosonic ghost system106

4.21 Orbifolds107

4.22 CFT on Other Surfaces of Euler Number Zero112

4.23 CFT on Higher-genus Riemann Surfaces116

Bibliography117

Exercises118

5 Scattering Amplitudes and Vertex Operators126

5.1 Physical Vertex Operators128

5.2 Calculation of Tree-level Tachyon Amplitudes130

5.2.1 The closed string130

5.2.2 The open string131

5.3 The One-loop Vacuum Amplitudes133

5.3.1 The torus134

5.3.2 The cylinder136

5.3.3 The Klein bottle138

5.3.4 The M？bius strip138

5.3.5 Tadpole cancellation139

5.3.6 UV structure and UV-IR correspondence140

Bibliography141

Exercises142

6 Strings in Background Fields144

6.1 The Nonlinearσ-model Approach144

6.2 The Quest for Conformal Invariance147

6.3 Linear Dilaton and Strings in D＜26 Dimensions149

6.4 T-duality in Nontrivial Backgrounds151

Bibliography151

Exercises152

7 Superstrings and Supersymmetry155

7.1 N=(1,1)2 World-sheet Superconformal Symmetry155

7.2 Closed(Type-Ⅱ)Superstrings157

7.2.1 Massless R-R states159

7.3 Type-Ⅰ Superstrings162

7.4 Heterotic Superstrings165

7.5 Superstring Vertex Operators168

7.6 One-loop Superstring Vacuum Amplitudes170

7.6.1 The type-IIA/B superstring170

7.6.2 The heterotic superstring171

7.6.3 The type-Ⅰsuperstring171

7.7 Closed Superstrings and T-duality174

7.7.1 The type-Ⅱ string theories174

7.7.2 The heterotic string175

7.8 Supersymmetric Effective Actions175

7.9 Anomalies176

Bibliography182

Exercises183

8 D-branes187

8.1 Antisymmetric Tensors and p-branes187

8.2 Open Strings and T-duality188

8.3 D-branes191

8.4 D-branes and R-R Charges193

8.4.1 D-instantons196

8.5 D-brane Efiective Actions197

8.5.1 The Dirac-Born-Infeld action197

8.5.2 Anomaly-related terms199

8.6 Multiple Branes and Nonabelian Symmetry200

8.7 T-duality and Orientifolds201

8.8 D-branes as Supergravity Solitons205

8.8.1 The supergravity solutions205

8.8.2 Horizons and singularities207

8.8.3 The extremal branes and their near-horizongeometry208

8.9 NS5-branes211

Bibliography213

Exercises213

9 Compactifications and Supersymmetry Breaking219

9.1 Narain Compactifications219

9.2 World-sheet versus Space-time Supersymmetry223

9.2.1 N=14 space-time supersymmetry225

9.2.2 N=24 space-time supersymmetry226

9.3 Orbifold Reduction of Supersymmetry228

9.4 A Heterotic Orbifold with N=24 Supersymmetry231

9.5 Spontaneous Supersymmetry Breaking235

9.6 A Heterotic N=14 Orbifold and Chirality in Four Dimensions237

9.7 Calabi-Yau Manifolds239

9.7.1 Holonomy241

9.7.2 Consequences of SU(3)holonomy242

9.7.3 The CY moduli space243

9.8 N=14 Heterotic Compactifications245

9.8.1 The low-energy N=14 heterotic spectrum246

9.9 K3 Compactification of the Type-Ⅱ String247

9.10 N=26 Orbifolds of the Type-Ⅱ String250

9.11 CY Compactifications of Type-Ⅱ Strings252

9.12 Mirror Symmetry253

9.13 Absence of Continuous Global Symmetries255

9.14 Orientifolds256

9.14.1 K3 orientifolds257

9.14.2 The Klein bottle amplitude258

9.14.3 D-branes on T4/Z2260

9.14.4 The cylinder amplitude263

9.14.5 The M？bius strip amplitude265

9.14.6 Tadpole cancellation266

9.14.7 The open string spectrum267

9.15 D-branes at Orbifold Singularities268

9.16 Magnetized Compactifications and Intersecting Branes271

9.16.1 Open strings in an internal magneticfield272

9.16.2 Intersecting branes277

9.16.3 Intersecting D6-branes278

9.17 Where is the Standard Model?280

9.17.1 The heterotic string280

9.17.2 Type-Ⅱ string theory282

9.17.3 The type-Ⅰstring283

9.18 Unification284

Bibliography286

Exercises287

10 Loop Corrections to String Effective Couplings294

10.1 Calculation of Heterotic Gauge Thresholds296

10.2 On-shell Infrared Regularization301

10.2.1 Evaluation of the threshold303

10.3 Heterotic Gravitational Thresholds304

10.4 One-loop Fayet-Iliopoulos Terms305

10.5 N=1,24 Examples of Threshold Corrections309

10.6 N=24 Universality of Thresholds312

10.7 Unification Revisited315

Bibliography317

Exercises317

11 Duality Connections and Nonperturbative Effects320

11.1 Perturbative Connections322

11.2 BPS States and BPS Bounds323

11.3 Nonrenormalization Theorems and BPS-saturated Couplings325

11.4 Type-ⅡA versus M-theory328

11.5 Self-duality of the Type-ⅡB String331

11.6 U-duality of Type-Ⅱ String Theory334

11.6.1 U-duality and bound states336

11.7 Heterotic/Type Ⅰ Duality in Ten Dimensions336

11.7.1 The type-Ⅰ D1-string339

11.7.2 The type-Ⅰ D5-brane341

11.7.3 Further consistency checks343

11.8 M-theory and the E8×E8 Heterotic String344

11.8.1 Unification at strong heterotic coupling347

11.9 Heterotic/Type Ⅱ Duality in Six Dimensions348

11.9.1 Gauge synmetry enhancementand singular K3 surfaces352

11.9.2 Heterotic/type Ⅱ duality in four dimensions355

11.10 Conifold Singularities and Conifold Transitions356

Bibliography362

Exercises363

12 Black Holes and Entropy in String Theory369

12.1 A Brief History369

12.2 The Strategy370

12.3 Black-hole Thermodynamics371

12.3.1 The Euclidean continuation372

12.3.2 Hawking evaporation and greybody factors374

12.4 The Information Problem and the Holographic Hypothesis375

12.5 Fire-dimensional Extremal Charged Black Holes377

12.6 Five-dimensional Nonextremal RN Black Holes379

12.7 The Near-horizon Region381

12.8 Semiclassical Derivation of the Hawking Rate383

12.9 The Microscopic Realization386

12.9.1 Theworld-volume theory of the bound state387

12.9.2 The low-energy SCFT of the Dr-D5 bound state389

12.9.3 Microscopic calculation of the entropy391

12.9.4 Microscopic derivation of Hawking evaporation rates394

12.10 Epilogue396

Bibliography398

Exercises399

13 The Bulk/Boundary Correspondence403

13.1 Large-N Gauge Theories and String Theory405

13.2 The Decoupling Principle408

13.3 The Near-horizon Limit409

13.4 Elements of the Correspondence410

13.5 Bulk Fields and Boundary Operators413

13.6 Holography416

13.7 Testing the AdS5/CFT4 Correspondence417

13.7.1 The chiral spectrum of N=4 gaugetheory418

13.7.2 Matching to the string theory spectrum420

13.7.3 N=8 five-dimensional gauged supergravity422

13.7.4 Protected correlation functions and anomalies422

13.8 Correlation Functions424

13.8.1 Two-pointfunctions425

13.8.2 Three-pointfunctions427

13.8.3 The gravitational action and the conformal anomaly428

13.9 Wilson Loops433

13.10 AdS5/CFT4 Correspondence at Finite Temperature436

13.10.1 N=4 super yang-Mills theory atfinite temperature436

13.10.2 The near-horizon limit of black D3-branes438

13.10.3 Finite-volume and large-Nphase transitions440

13.10.4 Thermal holographic physics443

13.10.5 Spatial Wilson loops in(a version of)QCD3444

13.10.6 The glueball mass spectrum446

13.11 AdS3/CFT2 Correspondence447

13.11.1 The greybodyfuctors revisited450

13.12 The Holographic Renormalization Group450

13.12.1 Perturbations of the CFT4451

13.12.2 Domain walls and flow equations452

13.12.3 A RG flow preserving N=1 supersymmetry454

13.13 The Randall-Sundrum Geometry456

13.13.1 An alternative to compactification459

Bibliography462

Exercises463

14 String Theory and Matrix Models470

14.1 M(atrix) Theory471

14.1.1 Membrane quantization471

14.1.2 Type-ⅡA Do-branes and DLCQ473

14.1.3 Gravitons and branes in M(atrix)theory476

14.1.4 The two-graviton interaction from M(atrix) theory477

14.2 Matrix Models and D=1 Bosonic String Theory479

14.2.1 The continuum limit481

14.2.2 Solving the matrix model482

14.2.3 The double-scaling limit485

14.2.4 The free-fermion picture487

14.3 Matrix Description of D=2 String Theory488

14.3.1 Matrix quantum mechanics and free fermions on the line490

14.3.2 The continuum limit492

14.3.3 The double-scaling limit494

14.3.4 D-particles,tachyons,and holography496

Bibliography498

Exercises498

Appendix A Two-dimensional Complex Geometry503

Appendix B Differential Forms505

Appendix C Theta and Other Elliptic Functions507

C.1 v and Related Functions507

C.2 The Weierstrass Function510

C.3 Modular Forms510

C.4 Poisson Resummation512

Appendix D Toroidal Lattice Sums513

Appendix E Toroidal Kaluza-Klein Reduction516

Appendix F The Reissner-Nordstr？m Black Hole519

Appendix G Electric-magnetic Duality in D=4522

Appendix H Supersymmetric Actions in Ten and Eleven Dimensions525

H.1 The N=111 Supergravity526

H.2 Type-ⅡA Supergravity527

H.3 Type-ⅡB Supergravity528

H.4 Type-Ⅱ Supergravities：The Democratic Formulation529

H.5 N=110 Supersymmetry530

Appendix Ⅰ N=1,2,Four-dimensional Supergravity Coupled to Matter533

I.1 N=14 Supergravity533

I.2N=24 Supergravity535

Appendix J BPS Multiplets in Four Dimensions537

Appendix K The Geometry of Anti-de Sitter Space541

K.1 The Minkowski Signature AdS541

K.2 Euclidean AdS544

K.3 The Conformal Structure of Flat Space546

K.4 Fields in AdS548

K.4.1 The wave equation in Poincaré coordinates549

K.4.2 The bulk-boundary propagator550

K.4.3 The bulk-to-bulk propagator551

Bibliography553

Index575