String Field Theory; Modern Introduction

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Preface Organization Acknowledgements References Contents About the Author 1 Introduction 1.1 Strings, a Distinguished Theory 1.2 String Theory 1.2.1 Properties Worldsheet CFT Spectrum 1.2.2 Classification of Superstring Theories 1.2.3 Interactions Worldsheet and Riemann Surfaces Amplitudes Divergences and Feynman Graphs 1.3 String Field Theory 1.3.1 From the Worldsheet to Field Theory 1.3.2 String Field Action 1.3.3 Expression with Spacetime Fields 1.3.4 Applications 1.4 Suggested Readings References Part I Worldsheet Theory 2 Worldsheet Path Integral: Vacuum Amplitudes 2.1 Worldsheet Action and Symmetries 2.2 Path Integral 2.3 Faddeev–Popov Gauge Fixing 2.3.1 Metrics on Riemann Surfaces 2.3.2 Reparametrizations and Analysis of P1 Teichmüller Deformations Conformal Killing Vectors 2.3.3 Weyl Transformations and Quantum Anomalies 2.3.4 Ambiguities, Ultralocality and Cosmological Constant 2.3.5 Gauge Fixed Path Integral 2.4 Ghost Action 2.4.1 Actions and Equations of Motion 2.4.2 Weyl Ghost 2.4.3 Zero-Modes 2.5 Normalization 2.6 Summary 2.7 Suggested Readings References 3 Worldsheet Path Integral: Scattering Amplitudes 3.1 Scattering Amplitudes on Moduli Space 3.1.1 Vertex Operators and Path Integral 3.1.2 Gauge Fixing: General Case 3.1.3 Gauge Fixing: 2-Point Amplitude Review of the Problem Computation of the Amplitude Expression with Ghosts 3.2 BRST Quantization 3.2.1 BRST Symmetry 3.2.2 BRST Cohomology and Physical States 3.3 Summary 3.4 Suggested Readings References 4 Worldsheet Path Integral: Complex Coordinates 4.1 Geometry of Complex Manifolds 4.2 Complex Representation of Path Integral 4.3 Summary 4.4 Suggested Readings References 5 Conformal Symmetry in D Dimensions 5.1 CFT on a General Manifold 5.2 CFT on Minkowski Space 5.3 Suggested Readings References 6 Conformal Field Theory on the Plane 6.1 The Riemann Sphere 6.1.1 Map to the Complex Plane 6.1.2 Relation to the Cylinder: String Theory 6.2 Classical CFTs 6.2.1 Witt Conformal Algebra 6.2.2 PSL(2, C) Conformal Group 6.2.3 Definition of a CFT 6.3 Quantum CFTs 6.3.1 Virasoro Algebra 6.3.2 Correlation Functions 6.4 Operator Formalism and Radial Quantization 6.4.1 Radial Ordering and Commutators 6.4.2 Operator Product Expansions 6.4.3 Hermitian and BPZ Conjugation 6.4.4 Mode Expansion 6.4.5 Hilbert Space State-Operator Correspondence Euclidean and BPZ Conjugations and Inner Products Verma Modules Normal Ordering 6.4.6 CFT on the Cylinder 6.5 Suggested Readings References 7 CFT Systems 7.1 Free Scalar 7.1.1 Covariant Action 7.1.2 Action on the Complex Plane 7.1.3 OPE 7.1.4 Mode Expansions 7.1.5 Commutators 7.1.6 Hilbert Space 7.1.7 Euclidean and BPZ Conjugates 7.2 First-Order bc Ghost System 7.2.1 Covariant Action 7.2.2 Action on the Complex Plane 7.2.3 OPE 7.2.4 Mode Expansions 7.2.5 Commutators 7.2.6 Hilbert Space Energy Vacuum (Grassmann Odd) Energy Normal Ordering (Grassmann Odd) Structure of the Hilbert Space (Grassmann Odd) 7.2.7 Euclidean and BPZ Conjugates 7.2.8 Summary 7.3 Suggested Readings References 8 BRST Quantization 8.1 BRST for Reparametrization Invariance 8.2 BRST in the CFT Formalism 8.2.1 OPE 8.2.2 Mode Expansions 8.2.3 Commutators 8.3 BRST Cohomology: Two Flat Directions 8.3.1 Conditions on the States 8.3.2 Relative Cohomology Light-Cone Parametrization Reduced Cohomology Full Relative Cohomology 8.3.3 Absolute Cohomology, States and No-Ghost Theorem 8.3.4 Cohomology for Holomorphic and Anti-holomorphic Sectors 8.4 Summary 8.5 Suggested Readings References Part II String Field Theory 9 String Field 9.1 Field Functional 9.2 Field Expansion 9.3 Summary 9.4 Suggested Readings References 10 Free BRST String Field Theory 10.1 Classical Action for the Open String 10.1.1 Warm-Up: Point-Particle 10.1.2 Open String Action 10.1.3 Gauge Invariance 10.1.4 Siegel Gauge 10.2 Open String Field Expansion, Parity and Ghost Number 10.3 Path Integral Quantization 10.3.1 Tentative Faddeev–Popov Gauge Fixing 10.3.2 Tower of Ghosts 10.4 Spacetime Action 10.5 Closed String 10.6 Summary 10.7 Suggested Readings References 11 Introduction to Off-Shell String Theory 11.1 Motivations 11.1.1 3-Point Function 11.1.2 4-Point Function 11.2 Off-Shell States 11.3 Off-Shell Amplitudes 11.3.1 Amplitudes from the Marked Moduli Space 11.3.2 Local Coordinates 11.4 Suggested Readings References 12 Geometry of Moduli Spaces and Riemann Surfaces 12.1 Parametrization of P(g,n) 12.2 Tangent Space 12.3 Plumbing Fixture 12.3.1 Separating Case 12.3.2 Non-separating Case 12.3.3 Decomposition of Moduli Spaces and DegenerationLimit 12.3.4 Stubs 12.4 Summary 12.5 Suggested Readings References 13 Off-Shell Amplitudes 13.1 Cotangent Spaces and Amplitudes 13.1.1 Construction of Forms 13.1.2 Amplitudes and Surface States 13.2 Properties of Forms 13.2.1 Vanishing of Forms with Trivial Vectors 13.2.2 BRST Identity 13.3 Properties of Amplitudes 13.3.1 Restriction to hat P(g,n) 13.3.2 Consequences of the BRST Identity 13.4 Suggested Readings References 14 Amplitude Factorization and Feynman Diagrams 14.1 Amplitude Factorization 14.1.1 Separating Case 14.1.2 Non-separating Case 14.2 Feynman Diagrams and Feynman Rules 14.2.1 Feynman Graphs 14.2.2 Propagator 14.2.3 Fundamental Vertices Recursive Definition: Tree-Level Vertices Recursive Definition: General Vertices Other Vertices 14.2.4 Stubs 14.2.5 1PI Vertices 14.3 Properties of Fundamental Vertices 14.3.1 String Product 14.3.2 Feynman Graph Interpretation 14.4 Suggested Readings References 15 Closed String Field Theory 15.1 Closed String Field Expansion 15.2 Gauge Fixed Theory 15.2.1 Kinetic Term and Propagator 15.2.2 Interactions 15.2.3 Action 15.3 Classical Gauge Invariant Theory 15.4 BV Theory 15.5 1PI Theory 15.6 Suggested Readings References 16 Background Independence 16.1 The Concept of Background Independence 16.2 Problem Setup 16.3 Deformation of the CFT 16.4 Expansion of the Action 16.5 Relating the Equations of Motion 16.6 Idea of the Proof 16.7 Suggested Readings References 17 Superstring 17.1 Worldsheet Superstring Theory 17.1.1 Heterotic Worldsheet Beta–Gamma System 17.1.2 Hilbert Spaces 17.2 Off-Shell Superstring Amplitudes 17.2.1 Amplitudes 17.2.2 Factorization 17.2.3 Spurious Poles Origin Vertical Integration: Large Hilbert Space Vertical Integration: Small Hilbert Space 17.3 Superstring Field Theory 17.3.1 String Field and Propagator 17.3.2 Constraint Approach 17.3.3 Auxiliary Field Approach 17.3.4 Large Hilbert Space 17.4 Suggested Readings References 18 Momentum-Space SFT 18.1 General Form 18.2 Generalized Wick Rotation 18.3 Suggested Readings References A Conventions A.1 Coordinates A.2 Operators A.3 QFT A.4 Curved Space and Gravity A.5 List of Symbols B Summary of Important Formulas B.1 Complex Analysis B.2 QFT, Curved Spaces and Gravity B.2.1 Two Dimensions B.3 Conformal Field Theory B.3.1 Complex Plane B.3.2 General Properties B.3.3 Hermitian and BPZ Conjugations Hermitian BPZ B.3.4 Scalar Field B.3.5 Reparametrization Ghosts B.4 Bosonic String C Quantum Field Theory C.1 Path Integrals C.1.1 Integration Measure C.1.2 Field Redefinitions C.1.3 Zero-Modes C.2 BRST Quantization C.3 BV Formalism C.3.1 Properties of Gauge Algebra C.3.2 Classical BV C.3.3 Quantum BV C.4 Suggested Readings References Index