Graduate students typically enter into courses on string theory having little to no familiarity with the mathematical background so crucial to the discipline. As such, this book, based on lecture notes, edited and expanded, from the graduate course taught by the author at SISSA and BIMSA, places particular emphasis on said mathematical background. The target audience for the book includes students of both theoretical physics and mathematics. This explains the book’s "strange" style: on the one hand, it is highly didactic and explicit, with a host of examples for the physicists, but, in addition, there are also almost 100 separate technical boxes, appendices, and starred sections, in which matters discussed in the main text are put into a broader mathematical perspective, while deeper and more rigorous points of view (particularly those from the modern era) are presented. The boxes also serve to further shore up the reader’s understanding of the underlying math. In writing this book, the author’s goal was not to achieve any sort of definitive conciseness, opting instead for clarity and "completeness". To this end, several arguments are presented more than once from different viewpoints and in varying contexts.
Author(s): Sergio Cecotti
Series: Theoretical and Mathematical Physics
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 828
City: Cham
Tags: string theory, conformal field theory, M-theory, Superstring D-branes, Calabi-Yau manifolds, Superstring interactions, BRST quantization, Polyakov path integral
Preface
Introduction: Why String Theory?
Contents
Acronyms
Conventions and Symbols
General Conventions
Symbols
List of Off-Text Technical Boxes
Part I Preliminary Matters
1 Introducing Strings: The Polyakov Path Integral
1.1 Introduction
1.1.1 History and Cartoons
1.1.2 Why String Theory?
1.1.3 String Theories: Geometric Classification
1.2 Bosonic String: The Polyakov Action
1.3 Bosonic String: Light-Cone Quantization
1.3.1 Quantization in the Light Cone
1.3.2 Lorentz Invariance: Emergence of Gravity
1.4 Covariant Quantization á la Polyakov
1.4.1 World-Sheet Topologies. Non-orientable Σ's
1.4.2 Conformal Killing Vectors and Complex Automorphisms
1.4.3 The Fadeev–Popov Determinant
1.4.4 The Matter Sector
1.5 The Weyl Anomaly
1.5.1 Strings in Non-critical Dimensions
1.6 Ghost Zero-Modes: Aut(Σ) and WP Moduli Geometry
1.6.1 The Riemann–Roch Theorem
1.6.2 b Zero-Modes and the Moduli Space
1.7 The Superstring
1.8 Strings Moving in Curved Backgrounds
1.8.1 The Spacetime Effective Action
1.8.2 String Compactifications
1.9 Physical Amplitudes, S-Matrix, and Vertices
References
2 Review of 2d Conformal Field Theories
2.1 Spacetime Symmetries in QFT
2.2 Conformal Field Theory (CFT)
2.2.1 Conformal Automorphisms and Equivalences
2.2.2 Radial Quantization and the State-Operator Isomorphism
2.2.3 Operator Product Expansions (OPE)
2.3 CFT in 2d
2.3.1 Primary Fields
2.3.2 The Virasoro Algebra
2.3.3 Finite Conformal Transformation of T(z)
2.3.4 Representations on the Hilbert Space
2.3.5 Unitarity
2.3.6 General Chiral Algebras in 2d CFT
2.3.7 Partition Function and Modular Invariance
2.3.8 More on Correlation Functions. Normal Order
2.4 Example: The 2d Free Massless Scalar
2.5 Free SCFTs and Their Bosonization
2.5.1 b,c and β,γ Systems
2.5.2 Anomalous U(1) Current (``Ghost Number'')
2.5.3 Fermi/Bose Sea States
2.5.4 The U(1) Stress Tensor and Its Bosonization
2.5.5 Riemann–Roch and Bosonization: The Linear Dilaton CFT
2.5.6 Bosonization of β, γ: The c=-2 System
2.5.7 The Picture Charge
2.6 Inclusion of Boundaries: Non-orientable Surfaces
2.7 KaČ–Moody and Current Algebras
2.7.1 KaČ–Moody Algebras
2.7.2 The Sugawara Construction
2.7.3 Knizhnik–Zamolodchikov Equation
2.7.4 Simply Laced G at Level 1
2.7.5 Fermionic Realization of the Current Algebra
2.8 (1, 1) Superconformal Algebra
2.8.1 Primary Superfields
2.8.2 Ramond and Neveu–Schwarz Sectors
2.8.3 SCFT State-Operator Correspondence
2.8.4 Example: The Free SCFT
2.9 SO(2n) Current Algebra at Level 1 and Lattices
2.9.1 The SO(d-1,1) World-Sheet Current Algebra
2.9.2 Bosonization of the SO(2N) Current Algebra
2.9.3 Spin(8) Triality and Refermionization
2.10 On Classification of 2d Superconformal Algebras
2.10.1 Classification of 2d Superconformal Algebras
Appendix 1: Witten's Non-abelian 2d Bosonization
Appendix 2: Valued Graphs, Affine Lie Algebras, McKay Correspondence
References
Part II Constructing Superstring Theory
3 Spectrum, Vertices, and BRST Quantization
3.1 The Superstring Lorentz Current Algebra
3.2 The Physical Spectrum: Light-Cone Gauge
3.3 Old Covariant Quantization
3.4 OCQ: Physical Conditions Versus 2d Superfields
3.5 BRST Invariance: Generalities
3.6 BRST Quantization of the Bosonic String
3.7 BRST Quantization of the Superstring
3.7.1 Q-Homotopies: Picture Changing
3.7.2 BRST Cohomology in Operator Space: Vertices
3.7.3 RR Vertices and a Perturbative Theorem
3.8 Spacetime Supersymmetry
3.8.1 Supersymmetry Ward Identities: Absence of Tadpoles
3.9 Open Strings: Chan–Paton Degrees of Freedom
Appendix: Details on the No-Ghost Theorem
References
4 Bosonic String Amplitudes
4.1 Path Integrals for Non-compact Scalars
4.1.1 Scalar Amplitudes on World-Sheets with χ0
4.2 Amplitudes for the b, c CFT
4.3 The Veneziano Amplitude
4.4 Chan–Paton Labels and Gauge Interactions
4.5 Closed String Tree-Level Amplitudes
4.5.1 Closed String Amplitudes on the Disk and mathbbRP2
4.6 One-Loop Amplitudes: The Torus
4.7 One-Loop: The Cylinder
4.8 Boundary and Cross-Cap States
4.9 One-Loop: Klein Bottle and Möbius Strip
References
5 10d Superstring Theories
5.1 2d Global Gravitational Anomalies
5.2 Consistent Closed Superstring Theories in 10d
5.3 Consistent Unoriented and Open Superstrings
5.4 2d Fermionic Path Integrals
5.5 Modular Invariance in Type II
5.6 Divergences and Tadpoles in Type I Theories
5.6.1 Consistency of SO(32) Type I
References
6 Bosonic String: T-Duality & D-Branes
6.1 Toroidal Compactifications in Field Theory
6.2 2d CFT of a Compact Scalar
6.3 Bosonization: Riemann Identities for Partition Functions
6.4 T-Duality in Closed Strings
6.4.1 T-Duality for a Compact Scalar
6.4.2 T-Duality on a General Background and Buscher Rules
6.4.3 Compactification of Several Dimensions
6.5 Narain Compactifications
6.5.1 The T-Duality Group
6.6 Abelian Orbifolds
6.6.1 Twisting Procedure
6.6.2 More on the Kosterlitz–Thouless Transition Point
6.7 Open Strings: Adding Wilson Lines
6.8 Open Bosonic String: T-Duality
6.9 D-Branes
6.9.1 D-Brane Action (Bosonic String)
6.10 T-Duality of Unoriented Strings: Orientifolds
References
7 The Heterotic String
7.1 Constructing String Models
7.2 The SO(32) and E8timesE8 Heterotic Strings in 10d
7.3 Non-supersymmetric Heterotic Strings in 10d
7.4 Heterotic Strings: The Bosonic Construction
7.5 Classification of Even Self-dual Lattices
7.6 SUSY Heterotic Strings in d=10 (Bosonic Form)
7.7 Toroidal Compactifications
7.7.1 Relation Between E8timesE8 and SO(32) Heterotic Strings
7.7.2 Example: Toroidal Compactification to Four Dimensions
7.8 Supersymmetry and BPS States
References
Part III Physics of Supersymmetric Strings
8 Low-Energy Effective Theories
8.1 Supergravity: a Quick Review
8.2 Non-Renormalization Theorems. BPS Objects
8.3 Supergravity in 11d
8.4 Type IIA Superstring: Low-Energy Effective Theory
8.5 Type IIB: Effective Low-Energy Theory
8.6 Type I Superstring: Low-Energy Effective Theory
8.7 Heterotic String
8.8 BPS Solutions
References
9 Anomalies and All That
9.1 Review of the Anomaly Polynomial Formalism
9.2 Anomaly Cancelation in 10d SUSY String Theories
9.3 Modular-Invariant Anomaly-Free
References
10 Superstring Amplitudes Non-Renormalization Theorems
10.1 Tree-Level Amplitudes
10.2 General Amplitudes
10.3 One-Loop Amplitudes
10.4 Non-Renormalization Theorems Again
References
11 Calabi–Yau Compactifications
11.1 Geometric Background
11.1.1 Mini-Review of Differential Geometry (DG)
11.1.2 Complex and Kähler Manifolds
11.1.3 Calabi–Yau Manifolds (CY)
11.1.4 Ultra-short Review of Kodaira–Spencer (KS) Theory
11.2 Superstrings on CY Manifolds: The World-Sheet Perspective
11.2.1 Calabi–Yau 2d σ-Models as (2,2) SCFTs
11.3 (2,2) SCFTs as Type II Backgrounds
11.4 Mirror Symmetry
11.5 Heterotic E8timesE8 on a Calabi–Yau 3-fold
11.6 Type II Compactified on a 3-CY: the Spacetime Perspective
11.7 Lightning Review of 4d mathcalN=2 Supergravity
11.8 The Low-Energy Theory of Type IIB on a 3-CY X
11.9 The Hypermultiplet Sector. c-Map
11.10 Global Aspects
References
Part IV Superstrings Beyond Weak Coupling
12 Superstring D-Branes
12.1 T-Duality in Type II Strings
12.2 T-Duality of Type I Strings: SUSY D-Branes
12.3 Relations Between Superstring Theories
12.4 D-Brane Tensions and RR Charges
12.5 D-Brane Actions
12.6 Supersymmetric Multi-brane Arrangements
12.6.1 Branes of Different Dimension Parallel to the Axes
12.6.2 The World-Volume Viewpoint: The #ND=4 System
12.6.3 Non-parallel Branes
12.7 BPS Bound States of Branes
12.7.1 F1-D1 Bound States
12.7.2 D0-Dp Bound States
12.8 D-Branes as Yang–Mills Instantons
References
13 SUSY Strings at Strong Coupling
13.1 Type IIB Strings at Strong Coupling: SL(2,mathbbZ) Duality
13.1.1 SL(2,mathbbZ) Duality
13.1.2 D3-Branes and Montonen–Olive Duality
13.2 U-Duality
13.3 IIA on K3 is Dual to Heterotic on T4
13.4 SO(32) Type I-Heterotic Duality
13.4.1 The Type I D5-Brane Versus the Heterotic NS5
13.5 Type IIA at Strong Coupling: M-Theory
13.6 M-Theory BPS Objects Versus IIA Branes
13.7 The E8timesE8 Heterotic String at Strong Coupling
13.8 IIA D8-Branes Versus M-Theory
13.9 The Big Picture: What Is String Theory?
References
14 Applications and Further Topics
14.1 Taub-NUT and GH Geometries
14.1.1 Half-BPS 6-Branes in M-Theory. Non-Abelian Gauge Symmetry
14.2 ADHM Construction Versus D-Branes
14.3 The Idea of F-Theory
14.3.1 Duality Between M- and F-Theory
14.4 Matrix Theory: A Proposal for M-Theory
14.4.1 The M-Theory Membrane
14.5 6d (2,0) SCFTs
14.5.1 Construction of 6d (2, 0) SCFT from IIB on mathbbC2/Γ
14.6 Quantum Physics of Black Holes
References
Index
