We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough. Niels Bohr Superstring theory has emerged as the most promising candidate for a quan­ tum theory of all known interactions. Superstrings apparently solve a problem that has defied solution for the past 50 years, namely the unification of the two great fundamental physical theories of the century, quantum field theory and general relativity. Superstring theory introduces an entirely new physical picture into theoretical physics and a new mathematics that has startled even the mathematicians. Ironically, although superstring theory is supposed to provide a unified field theory of the universe, the theory itself often seems like a confused jumble offolklore, random rules of thumb, and intuition. This is because the develop­ ment of superstring theory has been unlike that of any other theory, such as general relativity, which began with a geometry and an action and later evolved into a quantum theory. Superstring theory, by contrast, has been evolving backward for the past 20 years. It has a bizarre history, beginning with the purely accidental discovery of the quantum theory in 1968 by G. Veneziano and M. Suzuki. Thumbing through old math books, they stumbled by chance on the Beta function, written down in the last century by mathematician Leonhard Euler.

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I. First Quantization and Path Integrals.- 1 Path Integrals and Point Particles.- 1.1. Why Strings?.- 1.2. Historical Review of Gauge Theory.- 1.3. Path Integrals and Point Particles.- 1.4. Relativistic Point Particles.- 1.5. First and Second Quantization.- 1.6. Faddeev-Popov Quantization.- 1.7. Second Quantization.- 1.8. Harmonic Oscillators.- 1.9. Currents and Second Quantization.- 1.10. Summary.- References.- 2 Nambu-Goto Strings.- 2.1. Bosonic Strings.- 2.2. Gupta-Bleuler Quantization.- 2.3. Light Cone Quantization.- 2.4. BRST Quantization.- 2.5. Trees.- 2.6. From Path Integrals to Operators.- 2.7. Projective Invariance and Twists.- 2.8. Closed Strings.- 2.9. Ghost Elimination.- 2.10. Summary.- References.- 3 Superstrings.- 3.1. Supersymmetric Point Particles.- 3.2. Two-Dimensional Supersymmetry.- 3.3. Trees.- 3.4. Local 2D Supersymmetry H.- 3.5. Quantization.- 3.6. GSO Projection.- 3.7. Superstrings.- 3.8. Light Cone Quantization of the GS Action.- 3.9. Vertices and Trees.- 3.10. Summary.- References.- 4 Conformal Field Theory and Kac-Moody Algebras.- 4.1. Conformal Field Theory.- 4.2. Superconformal Field Theory.- 4.3. Spin Fields.- 4.4. Superconformal Ghosts.- 4.5. Fermion Vertex.- 4.6. Spinors and Trees.- 4.7. Kac-Moody Algebras.- 4.8. Supersymmetry.- 4.9. Summary.- References.- 5 Multiloops and Teichmüller Spaces.- 5.1. Unitarity.- 5.2. Single-Loop Amplitude.- 5.3. Harmonic Oscillators.- 5.4. Single-Loop Superstring Amplitudes.- 5.5. Closed Loops.- 5.6. Multiloop Amplitudes.- 5.7. Riemann Surfaces and Teichmüller Spaces.- 5.8. Conformal Anomaly.- 5.9. Superstrings.- 5.10. Determinants and Singularities.- 5.11. Moduli Space and Grassmannians.- 5.12. Summary.- References.- II. Second Quantization and the Search for Geometry.- 6 Light Cone Field Theory.- 6.1. Why String Field Theory?.- 6.2. Deriving Point Particle Field Theory.- 6.3. Light Cone Field Theory.- 6.4. Interactions.- 6.5. Neumann Function Method.- 6.6. Equivalence of the Scattering Amplitudes.- 6.7. Four-String Interaction.- 6.8. Superstring Field Theory.- 6.9. Summary.- References.- 7 BRST Field Theory.- 7.1. Covariant String Field Theory.- 7.2. BRST Field Theory.- 7.3. Gauge Fixing.- 7.4. Interactions.- 7.5. Axiomatic Formulation.- 7.6. Proof of Equivalence.- 7.7. Closed Strings and Superstrings.- 7.8. Summary.- References.- 8 Geometric String Field Theory.- 8.1. Why Geometry?.- 8.2. The String Group.- 8.3. Unified String Group.- 8.4. Representations of the USG.- 8.5. Ghost Sector and the Tangent Space.- 8.6. Connections and Covariant Derivatives.- 8.7. Geometric Derivation of the Action.- 8.8. The Interpolating Gauge.- 8.9. Closed Strings and Superstrings.- 8.10. Summary.- References.- III. Phenomenology and Model Building.- 9 Anomalies and the Atiyah-Singer Theorem.- 9.1. Beyond GUT Phenomenology.- 9.2. Anomalies and Feynman Diagrams.- 9.3. Anomalies in the Functional Formalism.- 9.4. Anomalies and Characteristic Classes.- 9.5. Dirac Index.- 9.6. Gravitational and Gauge Anomalies.- 9.7. Anomaly Cancellation in Strings.- 9.8. A Simple Proof of the Atiyah-Singer Index Theorem.- 9.9. Summary.- References.- 10 Heterotic Strings and Compactification.- 10.1. Compactification.- 10.2. The Heterotic String.- 10.3. Spectrum.- 10.4. Covariant and Fermionic Formulations.- 10.5. Trees.- 10.6. Single-Loop Amplitude.- 10.7. E8 and Kac-Moody Algebras.- 10.8. 10D Without Supersymmetry.- 10.9. Lorentzian Lattices.- 10.10. Summary.- References.- 11 Calabi-Yau Spaces and Orbifolds.- 11.1. Calabi-Yau Spaces.- 11.2. Review of de Rahm Cohomology.- 11.3. Cohomology and Homology.- 11.4. Kähler Manifolds.- 11.5. Embedding the Spin Connection.- 11.6. Fermion Generations.- 11.7. Wilson Lines.- 11.8. Orbifolds.- 1L9. Four-Dimensional Superstrings.- 11.10. Summary.- 11.11. Conclusion.- References.- A. 1. A Brief Introduction to Group Theory.- A.2. A Brief Introduction to General Relativity.- A.3. A Brief Introduction to the Theory of Forms.- A.4. A Brief Introduction to Supersymmetry.- A.5. A Brief Introduction to Supergravity.- A.6. Glossary of Terms.- A.7. Notation.- References.