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Strings, Conformal Fields, and M-Theory Second Edition Michio Kaku Strings, Conformal Fields, and M-Theory Second Edition With 48 Illustrations Springer Contents Preface vii Acknowledgments ix I Conformal Field Theory and Perturbation Theory 1 1 Introduction to Superstrings 3 1.1 Quantizing the Relativistic String 9 1.2 Scattering Amplitudes 17 1.3 Supersymmetry . 22 1.4 2D SUSY Versus 10D SUSY 25 1.5 Types of Strings 30 1.6 Summary 32 2 BPZ Bootstrap and Minimal Models 38 2.1 Conformal Symmetry in D Dimensions 38 2.2 Conformal Group in Two Dimensions 41 2.3 Representations of the Conformal Group 45 2.4 Fusion Rules and Correlations Function 47 2.5 Minimal Models 51 2.6 Fusion Rules for Minimal Models 58 2.7 Superconformal Minimal Series 60 2.8 Summary 64 3 WZW Model, Cosets, and Rational Conformal Field Theory 69 3.1 Compactification and the WZW Model 69 3.2 Frenkel-Kac Construction 75 xii -Contents 3.3 GKO Coset Construction 79 3.4 Conformal and Current Blocks 81 3.5 Racah Coefficients for Rational Conformal Field Theory 84 3.6 Summary 91 4 Modular Invariance and the A-D-E Classification 96 4.1 Dehn Twists 96 4.2 Free Fermion and Boson Characters 99 4.3 GSO and Supersymmetry 105 4.4 Minimal Model Characters 106 4.5 Affine Characters 108 4.6 A-D-E Classification 113 4.7 Higher Invariants and Simple Currents 116 4.8 Diagonalizing the Fusion Rules 119 4.9 RCFT: Finite Number of Primary Fields 122 4.10 Summary 125 5 N = 2 SUSY and Parafermions 130 5.1 Calabi-Yau Manifolds 130 5.2 N = 2 Superconformal Symmetry 137 5.3 N - 2 Minimal Series 141 5.4 N = 2 Minimal Models and Calabi-Yau Manifolds .... 145 5.5 Parafermions 148 5.6 Supersymmetric Coset Construction 152 5.7 Hermitian Spaces 156 5.8 Summary 159 6 Yang-Baxter Relation 166 6.1 Statistical Mechanics and Critical Exponents 166 6.2 One-Dimensional Ising Model 168 6.3 Two-Dimensional Ising Model 171 6.4 RSOS and Other Models 172 6.5 Yang-Baxter Relation 179 6.6 Solitons and the Yang-Baxter Equation 186 6.7 Summary 189 7 Toward a Classification of Conformal Field Theories 196 7.1 Feigin-Fuchs Free Fields 196 7.2 Free Field Realizations of Coset Theories 203 7.3 Landau-Ginzburg Potentials 206 7.4 N = 2 Chiral Rings 209 7.5 N = 2 Landau-Ginzburg and Catastrophe Theory 211 7.6 Zamolodchikov's c Theorem 220 "— Contents xiii . 7.7 A-D-E Classification of c = 1 Theories 221 7.8 Summary 225 8 Knot Theory and Quantum Groups 231 8.1 Chern-Simons Approach to Conformal Field Theory . ... 231 8.2 Elementary Knot Theory 236 8.3 Jones Polynomial and the Braid Group 240 8.4 Quantum Field Theory and Knot Invariants 243 8.5 Knots and Conformal Field Theory 248 8.6 New Knot Invariants from Physics 251 8.7 Knots and Quantum Groups 255 8.8 Hecke and Temperley-Lieb Algebras 263 8.9 Summary 267 II Nonperturbative Methods 273 9 String Field Theory 275 9.1 First Versus Second Quantization 275 9.2 Light Cone String Field Theory 279 9.3 Free BRST Action 285 9.4 Interacting BRST String Field Theory 289 9.5 Four-Point Amplitude 293 9.6 Superstring Field Theory 297 9.7 Picture Changing 300 9.8 Superstring Action 303 9.9 Summary 306 10 Nonpolynomial String Field Theory 313 10.1 Four-String Interaction 313 10.2 N-Sided Polyhedra 324 10.3 Nonpolynomial Action 327 10.4 Conformal Maps 331 10.5 Tadpoles 337 10.6 Summary 341 11 2D Gravity and Matrix Models 346 11.1 Exactly Solvable Strings 346 11.2 2D Gravity and KPZ 349 11.3 Matrix Models 353 11.4 Recursion Relations 357 11.5 KdVHierarchy 362 11.6 Multimatrix Models 368 11.7 D = 1 Matrix Models 372 11.8 Summary 380 xiv Contents 12 Topological Field Theory 385 12.1 Unbroken Phase of String Theory 385 12.2 Topology and Morse Theory 388 12.3 Sigma Models and Floer Theory 394 12.4 Cohomological Topological Field Theories 398 12.5 Correlation Functions 403 12.6 Topological Sigma Models 406 12.7 Topological 2D Gravity 408 12.8 Correlation Functions for 2D Topological Gravity 410 12.9 Virasoro Constraint, W-Algebras, and KP Hierarchies ... 414 12.10 Summary 420 13 Seiberg-Witten Theory 427 13.1 Introduction 427 13.2 Electric-Magnetic Duality 429 13.3 Holomorphic Potentials 430 13.4 N = 1 SUSY QCD 433 13.4.1 N{ < Nc 437 13.4.2 N( = Nc 437 13.4.3 N{ = Nc + 1 438 13.4.4 Nc + 2<N{<lNc 438 13.4.5 §WC < Wf < 3WC 439 13.4.6 N(>3NC 439 13.4.7 SO(NC) SUSY Gauge Theory 439 13.5 N = 2 SUSY Gauge Theory 440 13.6 SU(N) N = 2 SUSY Gauge Theory 448 13.7 Summary 452 14 M-Theory and Duality 457 14.1 Introduction 457 14.2 Unifying the Five Superstring Theories 458 14.3 T Duality 459 14.4 5 duality 462 14.4.1 Type IIA and M-Theory 462 14.4.2 TypellB 464 14.4.3 Type I Strings 466 14.5 BPSStates 468 14.6 Supersymmetry and p-Branes 470 14.7 Compactification 473 14.8 Example: D = 6 475 14.8.1 D = 6, W = (2, 2) Theory 476 14.8.2 D = 6, W = (l, 1) Theories 476 14.8.3 Deletions and Fibrations 479 14.9 F-Theory 479 14.10 Summary 481 Contents xv 15 D-Branes and CFT/ADS Duality 487 15.1 Solitons 487 15.2 Supermembrane Action 489 15.3 5-Branes and D-Branes 491 15.4 D-Brane Actions . 495 15.5 M(atrix)-Theory and Membranes 500 15.6 Black Holes 504 15.7 CFT/ADS Duality 506 15.8 Anti-de Sitter Space 511 15.9 AdSandQCD 514 15.10 Summary 518 15.11 Conclusion 522 Index 525.
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