AdS/CFT Correspondence

Table of Contents

1. Introduction
2. Historical Context and Maldacena’s Proposal
3. Anti-de Sitter (AdS) Space
4. Conformal Field Theory (CFT) Basics
5. Duality Statement: AdS/CFT
6. D3-Branes and AdS\(_5\) × S\(^5\)
7. \( \mathcal{N}=4 \) Super Yang–Mills Theory
8. Dictionary of the Duality
9. Matching Symmetries
10. Radial Direction as Energy Scale
11. Boundary Operators and Bulk Fields
12. Correlation Functions and Partition Functions
13. Holographic Principle
14. Strong/Weak Coupling Duality
15. Black Holes and Thermodynamics
16. Holographic Entanglement Entropy
17. AdS/CMT and Applications in Condensed Matter
18. Holographic QCD
19. Holographic Renormalization
20. Higher-Spin Generalizations
21. Beyond AdS: de Sitter and Flat Holography
22. Extensions and Other Dualities
23. Open Problems and Limitations
24. Mathematical Impact
25. Conclusion

1. Introduction

The AdS/CFT correspondence, also known as gauge/gravity duality, is a conjectured relationship between a gravitational theory in anti-de Sitter (AdS) space and a conformal field theory (CFT) defined on its boundary. Proposed by Juan Maldacena in 1997, it provides a non-perturbative definition of quantum gravity and a powerful tool for studying strongly coupled quantum field theories.

2. Historical Context and Maldacena’s Proposal

Maldacena’s insight came from analyzing low-energy limits of D3-branes in Type IIB string theory. The key idea was that the dynamics near the branes could be described by both:

• Supergravity in AdS\(_5\) × S\(^5\)
• \( \mathcal{N}=4 \) Super Yang–Mills theory in 4D

3. Anti-de Sitter (AdS) Space

AdS space is a maximally symmetric spacetime with constant negative curvature. For AdS\(_{d+1}\), the metric can be written as:

\[
ds^2 = \frac{R^2}{z^2} (dz^2 + \eta_{\mu\nu} dx^\mu dx^\nu)
\]

Here:

• \( z \): radial coordinate (bulk)
• \( x^\mu \): boundary coordinates

4. Conformal Field Theory (CFT) Basics

CFTs are invariant under conformal transformations. In 4D, \( \mathcal{N}=4 \) Super Yang–Mills is a well-known CFT with:

• SU(N) gauge symmetry
• 6 real scalars, 4 Majorana fermions
• Exact conformal symmetry at the quantum level

5. Duality Statement: AdS/CFT

The core statement:

\[
\text{Type IIB string theory on } \text{AdS}_5 \times S^5 \quad \equiv \quad \mathcal{N}=4 \text{ SYM in 4D}
\]

This is a holographic duality: gravity in (d+1) dimensions is equivalent to a QFT in d dimensions.

6. D3-Branes and AdS\(_5\) × S\(^5\)

D3-branes in Type IIB string theory:

• Source RR flux
• Worldvolume theory is \( \mathcal{N}=4 \) SYM
• Near-horizon limit gives AdS\(_5\) × S\(^5\)

7. \( \mathcal{N}=4 \) Super Yang–Mills Theory

This theory is:

• Conformal for all values of coupling
• Integrable in the planar limit
• Exhibits exact duality properties (S-duality, T-duality)

8. Dictionary of the Duality

The AdS/CFT dictionary relates:

• Bulk fields ↔ Boundary operators
• Bulk action ↔ Generating functional
• Mass of bulk scalar ↔ Scaling dimension \( \Delta \) of boundary operator:

\[
\Delta(\Delta – d) = m^2 R^2
\]

9. Matching Symmetries

• AdS\(_5\) isometries ↔ Conformal group SO(4,2)
• S\(^5\) isometries ↔ SU(4) R-symmetry
• SUSY matching: both sides preserve 32 supercharges

10. Radial Direction as Energy Scale

The AdS radial coordinate \( z \) maps to energy scale in the CFT:

• \( z \to 0 \): UV
• \( z \to \infty \): IR

This connects renormalization group (RG) flow with holography.

11. Boundary Operators and Bulk Fields

A bulk field \( \phi(z, x) \) near boundary behaves as:

\[
\phi(z, x) \sim z^{d – \Delta} \phi_0(x)
\]

\[
\left\langle e^{\int \phi_0(x) \mathcal{O}(x)} \right\rangle = Z_{\text{string}}[\phi \to \phi_0]
\]

12. Correlation Functions and Partition Functions

Field theory correlators are computed via bulk string theory:

\[
\left\langle \mathcal{O}1(x_1) \dots \mathcal{O}_n(x_n) \right\rangle = \frac{\delta^n Z{\text{string}}}{\delta \phi_0(x_1) \dots \delta \phi_0(x_n)}
\]

13. Holographic Principle

AdS/CFT is a realization of the holographic principle — the idea that a higher-dimensional gravity theory can be encoded by degrees of freedom on a lower-dimensional boundary.

14. Strong/Weak Coupling Duality

AdS/CFT relates:

• Weakly coupled gravity ↔ Strongly coupled gauge theory
• Enables study of QCD-like systems at strong coupling using classical gravity

15. Black Holes and Thermodynamics

Black holes in AdS correspond to thermal states in the CFT:

• Hawking–Page transition ↔ Confinement/deconfinement transition
• Holographic entropy matches Bekenstein–Hawking formula

16. Holographic Entanglement Entropy

Ryu–Takayanagi formula:

\[
S_A = \frac{\text{Area}(\gamma_A)}{4 G_N}
\]

Where \( \gamma_A \) is the minimal surface in AdS homologous to boundary region \( A \). Provides geometric interpretation of entanglement.

17. AdS/CMT and Applications in Condensed Matter

Applied to strongly correlated systems:

• High-\( T_c \) superconductivity
• Quantum criticality
• Non-Fermi liquids

Gravity duals provide insight into transport and thermodynamics.

18. Holographic QCD

Holographic models approximate QCD:

• Hard wall and soft wall models
• Chiral symmetry breaking
• Glueballs and mesons via bulk modes

19. Holographic Renormalization

Regularizes AdS divergences:

• Add boundary counterterms
• Extract finite correlators
• Captures RG flow and anomalies

20. Higher-Spin Generalizations

Extensions involve Vasiliev theory:

• Dual to vector models
• Challenge standard assumptions of AdS/CFT

21. Beyond AdS: de Sitter and Flat Holography

Attempts to extend holography to:

• de Sitter space (dS/CFT): cosmological settings
• Flat space holography: asymptotically flat spacetimes

Still under development.

22. Extensions and Other Dualities

AdS/CFT inspired other dualities:

• ABJM duality (AdS\(_4\)/CFT\(_3\))
• AdS\(_3\)/CFT\(_2\)
• AdS\(_7\)/CFT\(_6\)

23. Open Problems and Limitations

• Proving AdS/CFT rigorously
• Understanding stringy and quantum corrections
• Realistic QCD duals
• Extension to time-dependent and cosmological backgrounds

24. Mathematical Impact

AdS/CFT has influenced:

• Geometric analysis
• Representation theory
• Category theory and topological invariants

25. Conclusion

The AdS/CFT correspondence stands as one of the most profound insights in theoretical physics. It provides a non-perturbative definition of quantum gravity, a holographic view of spacetime, and a powerful toolkit to explore strongly coupled systems. Its implications continue to evolve, shaping the landscape of string theory, gauge theories, and quantum gravity research.