String Theory in a Nutshell Book

Preface xv
Abbreviations xvii

Chapter 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

Chapter 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

Chapter 3: Quantization of Bosonic Strings 28
3.1 Covariant Canonical Quantization 28
3.2 Light-cone Quantization 31
3.3 Spectrum of the 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

Chapter 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 of the 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 superconformal symmetry 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

Chapter 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 Möbius strip 138
5.3.5 Tadpole cancellation 139
5.3.6 UV structure and UV-IR correspondence 140
Bibliography 141
Exercises 142

Chapter 6: Strings in Background Fields 144
6.1 The Nonlinear ?-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

Chapter 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-I superstring 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

Chapter 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 Effective 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-horizon geometry 208
8.9 NS5-branes 211
Bibliography 213
Exercises 213

Chapter 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 Möbius 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 magnetic field 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-I string 283
9.18 Unification 284
Bibliography 286
Exercises 287

Chapter 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

Chapter 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 symmetry enhancement and 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

Chapter 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 Five-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 The world-volume theory of the bound state 387
12.9.2 The low-energy SCFT of the D1-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

Chapter 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 gauge theory 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-point functions 425
13.8.2 Three-point functions 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 at finite temperature 436
13.10.2 The near-horizon limit of black D3-branes 438
13.10.3 Finite-volume and large-N phase 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 greybody factors 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

Chapter 14: String Theory and Matrix Models 470
14.1 M(atrix) Theory 471
14.1.1 Membrane quantization 471
14.1.2 Type-IIA D0 -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 ? 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-Nordstro..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.2 N = 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