Preface ........................................................ xv
Abbreviations ................................................ xvii
1. Introduction ................................................ l
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 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
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
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 Mobius 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 Nonlinear σ-model Approach ...................... 144
6.2. The Quest for Conformal Invariance .................. 147
6.3. Linear Dilaton and Strings in D2 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-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
8. D-brains .................................................. 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 hranes and their near-horizon
geometry .................................... 208
8.9. NS5-branes .......................................... 211
Bibliography .............................................. 213
Exercises ................................................. 213
9. Compactiflcations and Supersymmetry Breaking .............. 219
9.1. Narain Compactiflcations ............................ 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 = I4 Heterotic Compactiflcations .................. 245
9.8.1 The low-energy N = 14 heterotic spectrum ...... 246
9.9. К3 Compactification of the Type-II String ........... 247
9.10. N = 26 Orbifolds of the Type-II String .............. 250
9.11. CY Compactiflcations 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. К3 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 Mbbius 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
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 ........... 294
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 x 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 К3
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 greyhody 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
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 junctions ......................... 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
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 В. Differential Forms ................................ 505
Appendix C. Theta and Other Elliptic Functions ................ 507
C.l. θ 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-Nordstrom 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 К. The Geometry of Anti-de Sitter Space .............. 541
K.l. 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 Poincare coordinates ......... 549
K.4.2. The bulk-boundary propagator ...................... 550
K.4.3. The bulk-to-bulk propagator ....................... 551
Bibliography .................................................. 553
Index ......................................................... 575
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