String Field Theory; Modern Introduction

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