Advanced AdS/CFT

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The initial drafts were prepared with the assistance of AI and have since been reviewed and edited by me. Although I have tried to ensure accuracy, some errors may remain. If you notice any significant mistakes, I would greatly appreciate your feedback.

Aim of the course

This is the advanced version of the AdS/CFT curriculum. It assumes substantially more background than AdS/CFT Foundations, the companion course designed for readers with lower prerequisites.

Advanced AdS/CFT is a graduate and research-level route through the AdS/CFT correspondence. It is written for readers who already know the basics of quantum field theory, general relativity, conformal field theory, and string theory, and who want a coherent modern path from the original duality to the tools used in current research.

The guiding equality is

$$Z_{\mathrm{CFT}}[\text{sources}] = Z_{\mathrm{QG}}[\text{asymptotic boundary conditions}] .$$

This is not merely a resemblance between gauge theory and gravity. It is a statement about two complete quantum descriptions. In the semiclassical limit, it becomes the practical prescription

$$Z_{\mathrm{QG}}[\phi\to J] \simeq \exp\!\left(-S^{\mathrm{ren}}_{E,\mathrm{bulk}}[\phi_{\mathrm{cl}}[J]]\right),$$

so that CFT correlators are obtained by differentiating a renormalized on-shell bulk action.

The repeated discipline of the course is to specify the observable, dictionary entry, boundary and interior conditions, approximation, renormalization scheme, and regime of validity. Classical gravity is a powerful limit of AdS/CFT, not the whole correspondence.

Course map

AdS/CFT is learned by translating between four languages: CFT data, bulk geometry and effective field theory, string/D-brane constructions, and quantum information. The practical dictionary is controlled by parameters such as $N$, $\lambda$, $L/\ell_s$, and $L^{d-1}/G_{d+1}$.

The course contains 73 lecture-note pages plus this landing page. The early modules build the conceptual spine: holography, large $N$, AdS geometry, the source/operator dictionary, and the canonical D3-brane example. The middle modules develop calculational tools: correlators, Witten diagrams, renormalization, Wilson loops, black holes, real-time response, transport, RG flows, finite density, and quantum matter. The later modules cover entanglement, reconstruction, black-hole information, examples beyond AdS$_5$/CFT$_4$, and a research toolkit.

Prerequisites

Subject	Useful prior knowledge	What the course reinforces
QFT	Path integrals, symmetries, Ward identities, perturbation theory	Sources, generating functionals, Kubo formulas, large-$N$ factorization
CFT	Primaries, OPEs, scaling dimensions, stress tensor, radial quantization	CFT data, thermal CFTs, modular Hamiltonians, entanglement
GR	Metrics, curvature, horizons, black-hole thermodynamics	AdS boundaries, black branes, Brown-York tensors, Einstein equations
String theory	Worldsheets, D-branes, open/closed strings, compactification basics	D3/M2/M5 examples, flux quantization, brane probes, stringy corrections
Quantum information	Entropy, relative entropy, purification, basic codes	RT/HRT, QES, JLMS, entanglement wedges, error correction

The main dictionary, before details

Boundary object	Bulk object	Leading dictionary
CFT state $\vert\Psi\rangle$	Quantum-gravity state with specified asymptotics	Semiclassical states become geometries
Scalar primary $\mathcal O$	Bulk scalar $\phi$	$m^2L^2=\Delta(\Delta-d)$
Conserved current $J^\mu$	Bulk gauge field $A_M$	$A^{(0)}_\mu$ sources $J^\mu$
Stress tensor $T^{\mu\nu}$	Bulk metric $g_{MN}$	$g^{(0)}_{\mu\nu}$ sources $T^{\mu\nu}$
Single-trace operator	Single-particle bulk excitation	Large-$N$ factorization
Multi-trace operator	Multi-particle state or changed boundary condition	Double-trace deformations give mixed boundary conditions
Thermal state	AdS black hole or black brane	Entropy from horizon area
Retarded correlator	Lorentzian perturbation problem	Infalling condition at future horizon
Wilson loop $W(C)$	String worldsheet ending on $C$	$\langle W(C)\rangle\sim e^{-S_{\mathrm{NG}}}$
Defect/flavor sector	Probe brane	Worldvolume fields compute defect/flavor observables
Entanglement entropy $S_A$	Extremal surface or QES	$S_A=\mathrm{Area}/4G_N+S_{\mathrm{bulk}}+\cdots$
RG scale	Radial direction	Boundary is UV; deeper bulk is IR, with caveats

A useful hierarchy is

$$\text{exact duality} \quad\longrightarrow\quad \text{large }N \quad\longrightarrow\quad \text{classical string theory} \quad\longrightarrow\quad \text{classical Einstein gravity}.$$

Each arrow throws away corrections. Large $N$ suppresses bulk loops. Large ‘t Hooft coupling suppresses stringy $\alpha'$ corrections. A sparse low-dimension spectrum is needed for an Einstein-like local bulk effective theory.

Complete table of contents

01. Orientation

• What Is AdS/CFT?
• Holography Before AdS/CFT
• The Four Languages
• Map of the Course

02. 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

03. Anti-de Sitter geometry

• AdS as a Spacetime
• Coordinate Systems and the Boundary
• Causal Structure and the Cylinder
• Geodesics, Redshift, and RG Scale

04. 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

05. The canonical example

• D3-Branes and Two Low-Energy Limits
• $\mathcal N=4$ Super-Yang-Mills
• Type IIB on AdS$_5\times S^5$
• Parameters and Regimes of Validity
• First Tests of the Correspondence

06. 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

07. Wilson loops, branes, and nonlocal observables

• Wilson Loops and Fundamental Strings
• Heavy-Quark Potential
• Baryon Vertices and Wrapped Branes
• Defects, Interfaces, and Probe Branes

08. Finite temperature and black holes

• Thermal Field Theory and Euclidean Time
• AdS-Schwarzschild and Black Branes
• Hawking-Page and Confinement
• Thermodynamics of $\mathcal N=4$ Plasma
• Retarded Green Functions and Quasinormal Modes

09. Transport, hydrodynamics, and plasma physics

• Linear Response and Kubo Formulas
• Holographic Conductivity
• Shear Viscosity and $\eta/s$
• 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
• AdS$_2$ 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 Curve, Islands, and Modern Lessons

14. Beyond the canonical duality

• AdS$_3$/CFT$_2$ and Brown-Henneaux
• M2-Branes, ABJM, and AdS$_4$/CFT$_3$
• 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

Suggested paths through the course

For a first serious pass, read

$$01\to02\to03\to04\to05\to06\to08\to12\to15 .$$

This path gives the conceptual and calculational spine: large $N$, AdS geometry, the GKPW prescription, the canonical example, correlators, black holes, entanglement, and the research toolkit.

For black holes and gravity, read

$$03\to04\to06\to08\to09\to12\to13\to15 .$$

For quantum matter and transport, read

$$04\to06\to08\to09\to10\to11\to15 .$$

For string theory and branes, read

$$02\to05\to07\to10\to14 .$$

For entanglement and quantum information, read

$$03\to04\to08\to12\to13 .$$

Conventions used throughout

Unless stated otherwise, the course uses the following notation.

Symbol	Meaning
$d$	Boundary spacetime dimension
$d+1$	Bulk spacetime dimension
$L$	AdS radius
$G_{d+1}$	Bulk Newton constant in $d+1$ dimensions
$z$	Poincaré radial coordinate, boundary at $z=0$
$r$	Alternative radial coordinate, often boundary at $r\to\infty$
$\mu,\nu$	Boundary indices
$M,N$	Bulk indices
$\Delta$	Scaling dimension of a CFT primary
$\lambda$	’t Hooft coupling $g_{\mathrm{YM}}^2N$
$C_T$	Stress-tensor two-point coefficient, often the invariant measure of large $N$

The standard Poincaré AdS metric is

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right), \qquad z>0 .$$

The boundary is at $z=0$. A cutoff surface $z=\epsilon$ regulates UV divergences in the CFT. Holographic renormalization removes divergent dependence on $\epsilon$ by adding local counterterms on that cutoff surface.

What counts as understanding AdS/CFT?

Understanding AdS/CFT means being able to perform controlled translations. Given a boundary quantity such as

$$\langle\mathcal O(x)\mathcal O(0)\rangle, \qquad G^R_{J_xJ_x}(\omega,k), \qquad \langle W(C)\rangle, \qquad S_A, \qquad \langle T_{\mu\nu}\rangle_T,$$

you should be able to identify the bulk problem, solve or approximate it, renormalize the result, and state which assumptions made the calculation possible.

A good holographic calculation usually has this anatomy:

1. Identify the CFT observable and ensemble.
2. Identify the dual bulk field, geometry, brane, string, or extremal surface.
3. Specify boundary conditions at the AdS boundary.
4. Specify interior or horizon conditions when needed.
5. Solve the classical, quantum, or numerical bulk problem.
6. Add counterterms or subtract universal divergences.
7. Extract the response, entropy, free energy, or correlator.
8. Check Ward identities, thermodynamics, scaling, and limits.
9. State the regime of validity.

The final step is where many wrong holographic claims are born. Never skip it.

Common misconceptions to avoid

“Classical gravity is AdS/CFT.”

Classical Einstein gravity is a limit of AdS/CFT. The full duality is a statement about a quantum theory of gravity or string/M-theory and a nongravitational quantum theory.

“The radial direction is literally the RG scale.”

The radial/scale relation is powerful, especially near the boundary, but too local a statement is gauge-dependent. Wilsonian holographic RG, Fefferman-Graham expansions, causal wedges, and entanglement wedges each refine the slogan differently.

“Every bottom-up model is an exact dual.”

Bottom-up models can isolate universal mechanisms, but they are not automatically UV-complete string duals. Their value depends on whether the question being asked is universal, effective, or model-dependent.

“Euclidean and Lorentzian calculations differ only by $t=-i\tau$.”

Real-time response requires causal prescriptions. In a black-hole background, retarded correlators are selected by infalling boundary conditions at the future horizon.

“Large $N$ guarantees Einstein gravity.”

Large $N$ gives factorization and suppresses loops, but an Einstein-like local bulk also requires a sparse low-dimension spectrum and a large gap to higher-spin or stringy states.

Exercises

Exercise 1: Exact statement or approximation?

Classify each statement as an exact duality statement, a large-$N$ statement, a large-coupling gravity statement, a phenomenological model statement, or false as written.

1. The CFT partition function with sources equals the bulk quantum-gravity partition function with corresponding boundary conditions.
2. Connected correlators of suitably normalized single-trace operators are suppressed at large $N$.
3. Every holographic theory has $\eta/s=1/(4\pi)$.
4. A scalar primary of dimension $\Delta$ maps semiclassically to a scalar field with $m^2L^2=\Delta(\Delta-d)$.
5. A five-dimensional Einstein-Maxwell-scalar model is automatically a UV-complete string dual.

Solution

1. Exact duality statement, when both sides are fully defined.
2. Large-$N$ statement.
3. False as written. The value $\eta/s=1/(4\pi)$ is a leading result for broad classes of two-derivative Einstein-gravity duals, and can be modified by higher-derivative terms, finite-coupling corrections, anisotropy, or other effects.
4. Large-coupling or semiclassical bulk statement.
5. False as written. Such a model may be a useful bottom-up effective theory, but it is not automatically UV-complete.

Exercise 2: Which corrections are suppressed?

In the AdS$_5$/CFT$_4$ example,

$$\frac{L^4}{\alpha'^2}\sim \lambda, \qquad \frac{L^3}{G_5}\sim N^2.$$

Explain what becomes small when $N\to\infty$ and what becomes small when $\lambda\to\infty$.

Solution

The ratio $L^3/G_5\sim N^2$ controls the size of the classical gravitational action in AdS units. Large $N$ suppresses bulk quantum loops and gives large-$N$ factorization in the CFT.

The ratio $L^2/\alpha'\sim\sqrt\lambda$ controls the separation between the AdS curvature radius and the string length. Large $\lambda$ suppresses $\alpha'$ corrections and makes massive string states heavy compared with the AdS scale.

Thus large $N$ suppresses quantum-gravity loop corrections, while large $\lambda$ suppresses stringy higher-derivative corrections.

Further reading

• Juan Maldacena, “The Large $N$ Limit of Superconformal Field Theories and Supergravity,” arXiv:hep-th/9711200.
• S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov, “Gauge Theory Correlators from Non-Critical String Theory,” arXiv:hep-th/9802109.
• Edward Witten, “Anti de Sitter Space and Holography,” arXiv:hep-th/9802150.
• Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, and Yaron Oz, “Large $N$ Field Theories, String Theory and Gravity,” arXiv:hep-th/9905111.
• Kostas Skenderis, “Lecture Notes on Holographic Renormalization,” arXiv:hep-th/0209067.
• Sean A. Hartnoll, Andrew Lucas, and Subir Sachdev, “Holographic Quantum Matter,” arXiv:1612.07324.
• Mukund Rangamani and Tadashi Takayanagi, Holographic Entanglement Entropy, arXiv:1609.01287.

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