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Guides
Introduction
AdS/CFT Courses & Lecture Series
AdS/CFT Dictionary
Boundary terms and Einstein-scalar system
Einstein–Maxwell–Dilaton (EMD)
AdS2/CFT1: JT gravity and the SYK model
AdS3/CFT2 Guide
Double holography and black hole information
General Relativity
Stationary solutions in AdS
Catalog of AdS black holes
Black hole perturbations and master equations
Quasinormal modes (QNMs)
Hawking–Page Transition
Action and boundary terms
Hamiltonian systems with constraints
AdS/CFT Foundations
Overview
1. Orientation
Why Holography?
What the Duality Claims
Regimes of Validity
Conventions and Units
Roadmap
2. CFT and Large N
QFT Data and Generating Functionals
Conformal Symmetry: The Minimum Needed
Radial Quantization and the Cylinder
Large N Gauge Theory
Single-Trace Operators and Factorization
N=4 Super-Yang-Mills
Thermal States and Density Matrices
3. Anti-de Sitter Geometry
AdS as a Spacetime
Coordinate Systems in AdS
The Conformal Boundary of AdS
Global AdS and the Cylinder
Isometries and Conformal Symmetry
Fields in AdS
Black Holes and Black Branes
Euclidean AdS and Thermal Circles
4. String Theory Origin
Open and Closed Strings
D-Branes for Field Theorists
D3-Branes: Two Descriptions
Near-Horizon Geometry
The Decoupling Limit
The Parameter Map
Why Classical Gravity Emerges
5. The Basic Dictionary
Statement of the Correspondence
The GKP/Witten Prescription
Scalar Two-Point Functions
The Mass-Dimension Relation
Alternate Quantization and the BF Bound
One-Point Functions and VEVs
Spin, Symmetry, and Conserved Currents
Stress Tensor and the Metric
6. Holographic Renormalization
Why Renormalization Is Needed
Near-Boundary Expansion
Counterterms and the Renormalized Action
One-Point Functions from Variation
Ward Identities and Anomalies
Radial Hamiltonian Viewpoint
Practical Recipe
7. Correlators and Probes
Witten Diagrams
Three-Point Functions and Bulk Couplings
Four-Point Functions and Bulk Locality
Heavy Operators and Geodesics
Wilson Loops
Probe Branes and Flavor
Bulk Causality and Boundary Consistency
8. Thermal and Real-Time Holography
Black Branes and Thermal CFTs
Hawking–Page transition
Euclidean Gravity and Free Energy
Real-Time Prescription
Retarded Green Functions
Quasinormal Modes
Hydrodynamics from Gravity
Shear Viscosity
9. Entanglement and Geometry
Entanglement Entropy in QFT
Ryu–Takayanagi formula
Covariant HRT formula
Entanglement Wedges
Relative Entropy and Linearized Gravity
Quantum Extremal Surfaces
Islands and Information
10. AdS3/CFT2
Why AdS3 Is Special
Brown–Henneaux central charge
BTZ Black Holes
Cardy formula and black-hole entropy
Virasoro Symmetry and Boundary Gravitons
Lessons for Higher-Dimensional Holography
11. Applications and Beyond
What Counts as a Holographic CFT?
Other AdS Backgrounds
RG Flows and Domain Walls
Confinement and Hard-Wall Models
Finite Density and Bulk Gauge Fields
Reissner–Nordström-AdS
AdS2 Throats and IR Criticality
Holographic Conductivity
Holographic Superconductors
Fermions and Spectral Functions
Bottom-Up Models
Limits of the Dictionary
12. Bulk Quantum Gravity from CFT
Bulk Effective Field Theory
Large N Factorization and Fock Space
Bulk Reconstruction
Quantum Error Correction
Black-Hole Information in AdS/CFT
Stringy and Quantum Corrections
Open Problems and Research Map
13. Appendices
Notation
Differential Geometry Cheatsheet
AdS Coordinate Systems
Variational Principles and Boundary Terms
Green Functions Cheatsheet
CFT Cheatsheet
String Theory Cheatsheet
Problem Sets
Bibliography
Advanced AdS/CFT
Overview
1. Orientation
What Is AdS/CFT?
Holography Before AdS/CFT
The Four Languages of AdS/CFT
Map of the Course
2. Black Holes, Large N, and Strings
Black-Hole Entropy and the Holographic Principle
Large-N Gauge Theory
Strings from Flux Tubes
D-Branes as Gauge Theory and Gravity
3. Anti-de Sitter Geometry
AdS as a Spacetime
Coordinate Systems and the Boundary
Causal Structure and the Cylinder
Geodesics, Redshift, and RG Scale
4. CFT Data and the Dictionary
CFT Data, Large N, and Single-Trace Operators
Sources, Operators, and Generating Functionals
Fields in AdS and the Mass-Dimension Relation
Gauge Fields, Currents, and the Graviton
The GKPW Prescription
5. The Canonical Example
D3-Branes and Two Low-Energy Limits
N=4 Super-Yang-Mills
Type IIB on AdS5 x S5
Parameters and Regimes of Validity
First Tests of the Correspondence
6. Correlators, Witten Diagrams, and Renormalization
Scalar Two-Point Functions
Holographic Renormalization
Three-Point Functions and Cubic Couplings
Witten Diagrams and Large-N Perturbation Theory
Real-Time Correlators and Infalling Boundary Conditions
7. Wilson Loops, Branes, and Nonlocal Observables
Wilson Loops and Fundamental Strings
Heavy Quark Potential
Baryon Vertices and Wrapped Branes
Defects, Interfaces, and Probe Branes
8. Finite Temperature and Black Holes
Thermal Field Theory and Euclidean Time
AdS-Schwarzschild and Black Branes
Hawking-Page Transition and Confinement
Thermodynamics of the N=4 Plasma
Retarded Green Functions and Quasinormal Modes
9. Transport, Hydrodynamics, and Plasma Physics
Linear Response and Kubo Formulas
Holographic Conductivity
Shear Viscosity and η/s
Fluid/Gravity Correspondence
Heavy-Ion Lessons and Limitations
10. RG Flows, Confinement, and QCD-like Duals
Relevant Deformations and Domain Walls
Holographic c-Theorems
Hard-Wall and Soft-Wall Models
Confinement, Wilson Loops, and Mass Gaps
Flavor Branes, Chiral Symmetry, and Mesons
11. Finite Density and Holographic Quantum Matter
Charged Black Holes and Chemical Potential
AdS2 Throats and Local Criticality
Holographic Fermions and Fermi Surfaces
Holographic Superconductors
Momentum Relaxation and Strange-Metal Transport
12. Entanglement, Geometry, and Bulk Reconstruction
Entanglement Entropy in QFT
Ryu-Takayanagi and HRT
Quantum Corrections, JLMS, and Replica Methods
Entanglement Wedge and Subregion Duality
Entanglement First Law and Einstein Equations
Quantum Error Correction and Bulk Locality
13. Black Hole Information and Quantum Gravity
Eternal Black Holes and the Thermofield Double
Chaos, Shockwaves, and the Butterfly Effect
Information Loss in AdS/CFT
Page Curves, Islands, and Modern Lessons
14. Beyond the Canonical Duality
AdS3/CFT2 and Brown-Henneaux
M2-Branes, ABJM, and AdS4/CFT3
M5-Branes and Six-Dimensional CFTs
Higher-Spin, Vector Models, and Sparse Spectra
Nonconformal Branes and Generalized Holography
Flat Space, de Sitter, and Open Problems
15. Computation and Research Toolkit
How to Compute a Holographic Observable
Dictionary Tables
Common Normalizations and Conventions
Numerical Holography and the DeTurck Method
Reading Guide to the Literature
Capstone Problems
Holographic Quantum Matter
Overview
1. Orientation
What Is Holographic Quantum Matter?
A Condensed-Matter Primer for Holographers
Large N, Horizons, and Model Status
2. Thermal quantum critical matter
Quantum Critical Matter
Black Branes and Thermodynamics
Retarded Green Functions and Horizons
Quantum-Critical Transport
3. Finite density and metallic transport
Chemical Potential and Charged Black Branes
AdS2 Throats and Local Criticality
EMD, Lifshitz, and Hyperscaling Violation
Metallic Transport Without Quasiparticles
Momentum Relaxation, Lattices, and Disorder
Strange Metals and Planckian Transport
4. Fermions, flavor, and compressible matter
Holographic Fermions and Spectral Functions
Electron Stars, Dirac Hair, and Luttinger Counts
Probe Branes, Flavor, and DBI Transport
5. Ordered and topological response
Holographic Superconductors and Superfluids
Spatially Modulated Phases and Competing Orders
Magnetic Fields, Hall Transport, and Anomalies
6. Diagnostics, dynamics, and experiments
Entanglement, Chaos, and Diffusion
Nonequilibrium Holographic Quantum Matter
Experimental Connections and Epistemic Limits
7. Toolkit
Computing Holographic Observables
Dictionary Tables and Common Normalizations
Numerical Holography and Open Problems
Black Hole Information
Overview
1. Foundations
Black Hole Entropy and the Holographic Principle
Hawking Radiation and Information Loss
Page Curve and Fine-Grained Entropy
Fast Scrambling and Hayden–Preskill Recovery
Complementarity, AMPS, and Firewalls
2. Holographic Entropy
The Ryu–Takayanagi Formula
HRT and Covariant Holographic Entropy
FLM, Generalized Entropy, and Bulk Entanglement
Quantum Extremal Surfaces
Rényi Entropy, Replicas, and Cosmic Branes
Bit Threads and Entropy Inequalities
3. Entanglement Wedge and Reconstruction
Causal Wedge and HKLL Reconstruction
Entanglement Wedge, JLMS, and Relative Entropy
Entanglement Wedge Reconstruction
Holography as Quantum Error Correction
HaPPY Code and Tensor-Network Models
Operator-Algebra Quantum Error Correction
Modular Flow and Bulk Locality
4. Islands and Replica Wormholes
Evaporating Black Holes and Baths
The Island Formula
JT Gravity and the Schwarzian
The Page Curve in JT Gravity
Replica Wormholes
Double Holography
Islands in Higher Dimensions
What Is the Island?
5. Interiors, ER=EPR, and Complexity
Thermofield Double and Two-Sided Black Holes
ER=EPR and Traversable Wormholes
Black Hole Interior and State Dependence
Holographic Complexity: Volume and Action
Python's Lunch and Decoding Complexity
6. Puzzles and Research Frontiers
Factorization Puzzles and Euclidean Wormholes
Ensemble Averaging, Baby Universes, and JT Gravity
Holography of Information and Asymptotic Observables
Flat Space, de Sitter, and Realistic Evaporation
Open Problems in Black Hole Information
Modern CFT for AdS/CFT
Overview
1. Orientation
Why CFT Before AdS/CFT?
QFT Data and Observables
Euclidean and Lorentzian Signatures
Roadmap and References
2. RG Fixed Points
Wilsonian RG
The Stress-Tensor Trace
Critical Phenomena as CFTs
CFT Data: Spectrum and OPE Coefficients
3. Conformal Geometry
Conformal Transformations in d Dimensions
The Conformal Algebra
Conformal Compactification and the Cylinder
Embedding-Space Formalism
4. Operators, Sources and Ward Identities
Primaries and Descendants
Stress Tensor and Conserved Currents
Sources and Generating Functionals
Conformal Ward Identities
5. Correlation Functions
Scalar Two- and Three-Point Functions
Spinning Correlators
Four-Point Functions and Cross-Ratios
Lorentzian Correlators
6. Radial Quantization and Representations
Radial Quantization
Inner Products and Conjugation
Unitarity Bounds
Casimir and Conformal Families
7. OPE, Blocks and Bootstrap
The OPE as Operator Algebra
Conformal Blocks
Crossing and the Conformal Bootstrap
Lightcone Bootstrap and Large Spin
8. Essential 2D CFT
Complex Coordinates and Local Conformal Symmetry
Stress Tensor and the Virasoro Algebra
Virasoro Representations
Free Fields in 2D CFT
9. 2D CFT for Strings and AdS3
Minimal Models and the Ising CFT
Modular Invariance and the Cardy Formula
Current Algebras and WZW Models
Liouville and Noncompact CFT
10. Curved Space, Thermal CFT and Entanglement
Cylinder and Finite-Size Scaling
Weyl Anomaly
Thermal CFT
Entanglement and Modular Hamiltonian
11. Symmetry, Supersymmetry and SCFT
Global Symmetries and Currents
Supersymmetry Algebras
BPS Multiplets and Protected Data
Exactly Marginal Deformations and Conformal Manifolds
12. Large N CFT
Generalized Free Fields
Single-Trace and Multi-Trace Operators
Planar Expansion
Large-N OPE and Perturbation Theory
13. N = 4 SYM and the AdS/CFT Bridge
N=4 SYM Fields and Symmetries
Chiral Primaries and Protected Correlators
Wilson Loops, Defects, and Spin Chains
The Pre-AdS/CFT Dictionary
14. Appendices
Notation
Lie Algebras for CFT
Embedding-Space Formulas
Supersymmetry Cheatsheet
Problem Sets and Projects
Glossary
Bibliography and Reading Paths
String Theory I
String Theory I Introduction
Why Strings? Regge Behavior and Dual Resonance Models
The Relativistic Point Particle
The Nambu–Goto and Polyakov Actions
Worldsheet Symmetries, Boundary Conditions, and Conformal Gauge
Classical Rotating Strings and the Regge Slope
Mode Expansions and Canonical Quantization
Covariant Virasoro Constraints and Bosonic Spectra
Light-Cone Quantization and the Critical Dimension
Hagedorn Growth and String Thermodynamics
String Interactions and the Free Boson CFT
Stress Tensor, Primaries, and Conformal Transformations
Vertex Operators, OPEs, and the Virasoro Algebra
Radial Quantization and State–Operator Correspondence
Highest-Weight Modules, Null States, and the Kac Determinant
Minimal Models, Ward Identities, and Correlators
Bosonic Physical States and the No-Ghost Structure
The bc Ghost System and Ghost Zero Modes
The Polyakov Path Integral and Moduli
Weyl Anomaly, Liouville Theory, and BRST Quantization
Tree-Level String Amplitudes from Worldsheet Correlators
The Virasoro–Shapiro Amplitude and Factorization
Massless Closed-String Vertices and Gauge Invariance
The Low-Energy Effective Action
Sigma-Model Beta Functions and the Linear Dilaton
Worldsheet Supersymmetry and the NSR Action
Superconformal Algebra and NS/R Sectors
The Open Superstring Spectrum
Ten-Dimensional Spinors and Ramond Ground States
The GSO Projection and Open-String Supersymmetry
Closed Superstring Sectors and Type II Theories
Bosonization, Spin Fields, and Superghosts
NSR Vertex Operators and Picture Changing
Type I Strings, Orientifolds, and Chan–Paton Factors
Closed-String T-Duality and Enhanced Symmetry
Open-String T-Duality and D-Branes
D-brane dynamics, tachyon condensation, and RR charges
String Theory II
String Theory II Introduction
Relativistic Particles, Branes, and the Birth of the String
The Polyakov Action, Gauge Symmetry, and Virasoro Constraints
Mode Expansions, Regge Trajectories, and Bosonic String Quantization
Worldsheet Supersymmetry and the RNS Action
The Superconformal Algebra and NS/R Sectors
The RNS Spectrum and the GSO Projection
Bosonization, Spin Fields, and Spacetime Supersymmetry
Ghosts, Superghosts, and BRST Quantization
Superstring Vertex Operators and Pictures
Tree Amplitudes, Chan–Paton Factors, and Gauge Theory from Open Strings
Low-Energy Limits: DBI, Type I, and Type II Supergravity
Physical States and Type II Superstring Spectra
Unoriented Strings, Orientifolds, and Type I Theory
One-Loop Strings, Tori, and Modular Invariance
Circle Compactification and T-Duality
Thermal Strings and the Hagedorn Transition
Open Strings, T-Duality, and the Emergence of D-Branes
The DBI Action and D-Brane Worldvolume Fields
Open/Closed Duality and D-Brane Scattering
Type II T-Duality, RR Fields, and Preserved Supersymmetry
D-Brane Charges, BPS Bounds, and Dirac Quantization
D-Brane Interactions and the Annulus Amplitude
Intersecting Branes and Preserved Supersymmetry
Worldvolume Flux, BIons, and F/D Bound States
Type IIB S-Duality, (p,q) Strings, and the D3-Brane
Supergravity p-Branes, NS5-Branes, and Throat Geometries
M-Theory Origin of Type IIA Branes
Near-Extremal D3-Branes and Entropy
Open String Field Theory and Tachyon Condensation
Absorption by D-Branes and Two-Point Functions
The Decoupling Limit and the AdS/CFT Dictionary
Holographic Correlators and the Breitenlohner–Freedman Bound
Holographic Anomalies and the Central Charges of N=4 SYM
Calabi–Yau Compactification and Moduli
AdS5 × T¹,¹ and the Klebanov–Witten theory
Uncategorized
Finite-temperature QFT
N=4 SYM Guide
ABJM Learning Guide
Polyakov action
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Jie Ren (
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