Ⅰ First Quantization and Path Integrals 1
1 Path Integrals and Point Particles 3
1.1 Why Strings? 3
1.2 Historical Review of Gauge Theory 7
1.3 Path Integrals and Point Particles 18
1.4 Relativistic Point Particles 25
1.5 First and Second Quantization 28
1.6 Faddeev-Popov Quantization 30
1.7 Second Quantization 34
1.8 Harmonic Oscillators 37
1.9 Currents and Second Quantization 40
1.10 Summary 44
References 47
2 Nambu-Goto Strings 49
2.1 Bosonic Strings 49
2.2 Gupta-Bleuler Quantization 60
2.3 Light Cone Quantization 67
2.4 BRST Quantization 70
2.5 Trees 72
2.6 From Path Integrals to Operators 78
2.7 Projective Invariance and Twists 84
2.8 Closed Strings 87
2.9 Ghost Elimination 90
2.10 Summary 95
References 99
3 Superstrings 101
3.1 Supersymmetric Point Particles 101
3.2 Two-Dimensional Supersymmetry 104
3.3 Trees 111
3.4 Local Two-Dimensional Supersymmetry 117
3.5 Quantization 119
3.6 GSO Projection 123
3.7 Superstrings 126
3.8 Light Cone Quantization of the GS Action 128
3.9 Vertices and Trees 134
3.10 Summary 136
References 139
4 Conformal Field Theory and Kac-Moody Algebras 141
4.1 Conformal Field Theory 141
4.2 Superconformal Field Theory 150
4.3 Spin Fields 155
4.4 Superconformal Ghosts 158
4.5 Fermion Vertex 165
4.6 Spinors and Trees 167
4.7 Kac-Moody Algebras 170
4.8 Supersymmetry 174
4.9 Summary 174
References 177
5 Multiloops and Teichmüller Spaces 178
5.1 Unitarity 178
5.2 Single-Loop Amplitude 181
5.3 Harmonic Oscillators 185
5.4 Single-Loop Superstring Amplitudes 192
5.5 Closed Loops 195
5.6 Multiloop Amplitudes 200
5.7 Riemann Surfaces and Teichmüller Spaces 210
5.8 Conformal Anomaly 217
5.9 Superstrings 221
5.10 Determinants and Singularities 224
5.11 Moduli Space and Grassmannians 226
5.12 Summary 238
References 242
Ⅱ Second Quantization and the Search for Geometry 245
6 Light Cone Field Theory 247
6.1 Why String Field Theory? 247
6.2 Deriving Point Particle Field Theory 250
6.3 Light Cone Field Theory 254
6.4 Interactions 261
6.5 Neumann Function Method 267
6.6 Equivalence of the Scattering Amplitudes 272
6.7 Four-String Interaction 275
6.8 Superstring Field Theory 280
6.9 Summary 286
References 290
7 BRST Field Theory 291
7.1 Covariant String Field Theory 291
7.2 BRST Field Theory 297
7.3 Gauge Fixing 300
7.4 Interactions 303
7.5 Witten's String Field Theory 308
7.6 Proof of Equivalence 311
7.7 Closed Strings and Superstrings 317
7.8 Summary 328
References 331
Ⅲ Phenomenology and Model Building 335
8 Anomalies and the Atiyah-Singer Theorem 337
8.1 Beyond GUT Phenomenology 337
8.2 Anomalies and Feynman Diagrams 341
8.3 Anomalies in the Functional Formalism 346
8.4 Anomalies and Characteristic Classes 348
8.5 Dirac Index 353
8.6 Gravitational and Gauge Anomalies 357
8.7 Anomaly Cancellation in Strings 366
8.8 Summary 368
References 372
9 Heterotic Strings and Compactification 373
9.1 Compactification 373
9.2 The Heterotic String 378
9.3 Spectrum 383
9.4 Covariant and Fermionic Formulations 386
9.5 Trees 388
9.6 Single-Loop Amplitude 391
9.7 E8 and Kac-Moody Algebras 395
9.8 Lorentzian Lattices 398
9.9 Summary 400
References 403
10 Calabi-Yau Spaces and Orbifolds 404
10.1 Calabi-Yau Spaces 404
10.2 Review of de Rahm Cohomology 409
10.3 Cohomology and Homology 413
10.4 K？hler Manifolds 419
10.5 Embedding the Spin Connection 426
10.6 Fermion Generations 428
10.7 Wilson Lines 432
10.8 Orbifolds 434
10.9 Four-Dimensional Superstrings 438
10.10 Summary 449
References 453
Ⅳ M-Theory 455
11 M-Theory and Duality 457
11.1 Introduction 457
11.2 Duality in Physics 458
11.3 Why Five String Theories? 460
11.4 T-Duality 462
11.5 S-Duality 465
11.5.1 Type IIA Theory 466
11.5.2 Type IIB Theory 469
11.5.3 M-Theory and Type IIB Theory 471
11.5.4 E8？E8 Heterotic String 473
11.5.5 Type Ⅰ Strings 473
11.6 Summary 476
References 480
12 Compactifications and BPS States 482
12.1 BPS States 482
12.2 Supersymmetry and P-Branes 484
12.3 Compactification 488
12.4 Example:D=6 490
12.4.1 D=6,N=(2,2)Theory 491
12.4.2 D=6,N=(1,1)Theories 494
12.4.3 M-Theory in D=7 496
12.5 Example:D=4,N=2 and D=6,N=1 497
12.6 Symmetry Enhancement and Tensionless Strings 499
12.7 F-Theory 501
12.8 Example:D=4 502
12.9 Summary 504
References 510
13 Solitons,D-Branes,and Black Holes 511
13.1 Solitons 511
13.2 Supermembrane Actions 513
13.3 Five-Brahe Action 516
13.4 D-Branes 517
13.5 D-Brahe Actions 521
13.6 M(atrix) Models and Membranes 525
13.7 Black Holes 532
13.8 Summary 537
13.9 Conclusion 542
References 544
Appendix 545
A.1 A Brief Introduction to Group Theory 545
A.2 A Brief Introduction to General Relativity 557
A.3 A Brief Introduction to the Theory of Forms 561
A.4 A Brief Introduction to Supersymmetry 566
A.5 A Brief Introduction to Supergravity 573
A.6 Notation 577
References 579
Index 581