String theory continues to progress at an astonishing rate, and this book brings the reader up to date with the latest developments and the most active areas of research in the field. Building on the foundations laid in his Introduction to Superstrings and M Theory, Professor Kaku discusses such topics as the classification of conformal string theories, knot theory, the Yang-Baxter relation, quantum groups, and the insights into 11-dimensional strings recently obtained from M-theory. New chapters discuss such topics as Seiberg- Witten theory, M theory and duality., and D-branes. Several chapters review the fundamentals of string theory, making the presentation of the material self-contained while keeping overlap with the earlier book to a minimum. This book conveys the vitality of the current research and places readers at its forefront. Preface vii Acknowledgments ix I Conformal Field Theory and Perturbation Theory 1(272) Introduction to Superstrings 3(35) Quantizing the Relativistic String 9(8) Scattering Amplitudes 17(5) Supersymmetry 22(3) 2D SUSY Versus 10D SUSY 25(5) Types of Strings 30(2) Summary 32(6) BPZ Bootstrap and Minimal Models 38(31) Conformal Symmetry in D Dimensions 38(3) Conformal Group in Two Dimensions 41(4) Representations of the Conformal Group 45(2) Fusion Rules and Correlations Function 47(4) Minimal Models 51(7) Fusion Rules for Minimal Models 58(2) Superconformal Minimal Series 60(4) Summary 64(5) WZW Model, Cosets, and Rational Conformal Field Theory 69(27) Compactification and the WZW Model 69(6) Frenkel--Kac Construction 75(4) GKO Coset Construction 79(2) Conformal and Current Blocks 81(3) Racah Coefficients for Rational Conformal Field Theory 84(7) Summary 91(5) Modular Invariance and the A--D--E Classification 96(34) Dehn Twists 96(3) Free Fermion and Boson Characters 99(6) GSO and Supersymmetry 105(1) Minimal Model Characters 106(2) Affine Characters 108(5) A--D--E Classification 113(3) Higher Invariants and Sample Currents 116(3) Diagonalizing the Fusion Rules 119(3) RCFT: Finite Number of Primary Fields 122(3) Summary 125(5) N = 2 SUSY and Parafermions 130(36) Calabi--Yau Manifolds 130(7) N = 2 Superconformal Symmetry 137(4) N = 2 Minimal Series 141(4) N = 2 Minimal Models and Calabi--Yau Manifolds 145(3) Parafermions 148(4) Supersymmetric Coset Construction 152(4) Hermitian Space 156(3) Summary 159(7) Yang--Baxter Relation 166(30) Statistical Mechanics and Critical Exponents 166(2) One-Dimensional Ising Model 168(3) Two-Dimensional Ising Model 171(1) RSOS and Other Models 172(7) Yang--Baxter Relation 179(7) Solitons and the Yang--Baxter Equation 186(3) Summary 189(7) Toward a Classification of Conformal Field Theories 196(35) Feigin--Fuchs Free Fields 196(7) Free Field Realizations of Coset Theories 203(3) Landau--Ginzburg Potentials 206(3) N = 2 Chiral Rings 209(2) N= 2 Landau--Ginzburg and Catastrophe Theory 211(9) Zamolodchikov's c Theorem 220(1) A--D--E Classification of c = 1 Theories 221(4) Summary 225(6) Knot Theory and Quantum Groups 231(42) Chern--Simons Approach to Conformal Field Thoery 231(5) Elementary Knot Theory 236(4) Jones Polynomial and the Braid Group 240(3) Quantum Field Theory and Knot Invariants 243(5) Knots and Conformal Field Theory 248(3) New Knot Invariants from Physics 251(4) Knots and Quantum Groups 255(8) Hecke and Temperley--Lieb Algebras 263(4) Summary 267(6) II Nonperturbative Methods 273(252) String Field Theory 275(38) First Versus Second Quantization 275(4) Light Cone String Field Theory 279(6) Free BRST Action 285(4) Interacting BRST String Field Theory 289(4) Four-Point Amplitude 293(4) Superstring Field Theory 297(3) Picture Changing 300(3) Superstring Action 303(3) Summary 306(7) Nonpolynomial String Field Theory 313(33) Four-String Interaction 313(11) N-Sided Polyhedra 324(3) Nonpolynomial Action 327(4) Conformal Maps 331(6) Tadpoles 337(4) Summary 341(5) 2D Gravity and Matrix Models 346(39) Exactly Solvable Strings 346(3) 2D Gravity and KPZ 349(4) Matrix Models 353(4) Recursion Relations 357(5) KdV Hierarchy 362(6) Multimatrix Models 368(4) D = 1 Matrix Models 372(8) Summary 380(5) Topological Field Theory 385(42) Unbroken Phase of String Theory 385(3) Topology and Morse Theory 388(6) Sigma Models and Floer Theory 394(4) Cohomological Topological Field Theories 398(5) Correlation Functions 403(3) Topological Sigma Models 406(2) Topological 2D Gravity 408(2) Correlation Functions for 2D Topological Gravity 410(4) Virasoro Constraint, W-Algebras, and KP Hierarchies 414(6) Summary 420(7) Seiberg--Witten Theory 427(30) Introduction 427(2) Electric-Magnetic Duality 429(1) Holomorphic Potentials 430(3) N = 1 SUSY QCD 433(7) Nf < Nc 437(1) Nf = Nc 437(1) Nf = Nc + 1 438(1) Nc + 2 ? Nf ? 3/2 Nc 438(1) 3/2Nc Lt; Nf < 3Nc 439(1) Nf ? 3Nc 439(1) SO(Nc) SUSY Gauge Theory 439(1) N = 2 SUSY Gauge Theory 440(8) SU (N) N = 2 SUSY Gauge Theory 448(4) Summary 452(5) M-Theory and Daulity 457(30) Introduction 457(1) Unifying the Five Superstring Theories 458(1) T Daulity 459(3) S dautlity 462(6) Type IIA and M-Theory 462(2) Type IIB 464(2) Type I Strings 466(2) BPS States 468(2) Supersymmetry and p-Branes 470(3) Compactification 473(2) Example: D= 6 475(4) D = 6, N = (2, 2) Theory 476(1) D = 6, N = (1, 1) Theories 476(3) Deletions and Fibrations 479(1) F-Theory 479(2) Summary 481(6) D-Branes and CFT/ADS Daulity 487(38) Solitions 487(2) Supermembrane Action 489(2) 5-Branes and D-Branes 491(4) D-Brane Actions 495(5) M(atrix)-Theory and Membranes 500(4) Black Holes 504(2) CFT/ADS Daulity 506(5) Anti-de Sitter Space 511(3) AdS and QCD 514(4) Summary 518(4) Conclusion 522(3) Index 525