Chapter 8: AdS/CFT Correspondence
The Anti-de Sitter/Conformal Field Theory correspondence, proposed by Juan Maldacena in 1997, is the most concrete realization of holography. It states that quantum gravity in a bulk spacetime is exactly equivalent to a non-gravitational quantum field theory on its boundary.
1. The Maldacena Conjecture
Consider $N$ coincident D3-branes in Type IIB string theory. At low energies, there are two equivalent descriptions:
- Open string perspective: $\mathcal{N} = 4$ Super Yang–Mills theory with gauge group $SU(N)$
- Closed string perspective: Type IIB supergravity on the near-horizon geometry $\text{AdS}_5 \times S^5$
The conjecture equates these two descriptions exactly:
$$\text{Type IIB on } \text{AdS}_5 \times S^5 \;=\; \mathcal{N}=4 \;\text{SYM with } SU(N)$$
The parameters are related by:
$$\frac{L^4}{l_s^4} = 4\pi g_s N = \lambda, \quad g_s = \frac{g_{\text{YM}}^2}{4\pi}$$
where $\lambda = g_{\text{YM}}^2 N$ is the ’t Hooft coupling
2. Anti-de Sitter Space
Anti-de Sitter space is the maximally symmetric solution of Einstein’s equations with negative cosmological constant. In the Poincaré patch:
$$ds^2 = \frac{L^2}{z^2}\!\left(-dt^2 + d\vec{x}^2 + dz^2\right)$$
The boundary is at $z \to 0$ and the deep interior (horizon) at $z \to \infty$. The isometry group of $\text{AdS}_{d+1}$ is $SO(2,d)$, which is precisely the conformal group in $d$ dimensions.
$$\text{Isom}(\text{AdS}_{d+1}) = SO(2,d) = \text{Conf}(\mathbb{R}^{1,d-1})$$
3. The AdS/CFT Dictionary
The central equation of AdS/CFT equates the bulk and boundary partition functions:
$$Z_{\text{bulk}}[\phi_0] = \left\langle \exp\!\left(\int d^d x\; \phi_0(x)\,\mathcal{O}(x)\right)\right\rangle_{\text{CFT}}$$
Each bulk field $\phi$ of mass $m$ maps to a boundary operator$\mathcal{O}$ of conformal dimension $\Delta$:
$$m^2 L^2 = \Delta(\Delta - d)$$
Bulk
- Graviton $g_{MN}$
- Scalar field $\phi$
- Gauge field $A_M$
- Black hole temperature
Boundary
- Stress tensor $T_{\mu\nu}$
- Scalar operator $\mathcal{O}$
- Conserved current $J_\mu$
- CFT temperature
4. Ryu–Takayanagi Formula
The entanglement entropy of a boundary region $A$ is computed holographically by the area of the minimal surface $\gamma_A$ in the bulk anchored to$\partial A$:
$$S_A = \frac{\text{Area}(\gamma_A)}{4G_N}$$
For an interval of length $l$ in a 2D CFT (dual to AdS$_3$), the geodesic length gives:
$$S = \frac{c}{3}\ln\!\left(\frac{l}{\epsilon}\right)$$
matching the CFT result exactly, with $c = 3L/(2G_3)$
5. Strong/Weak Duality
The most powerful aspect of AdS/CFT is that it is a strong/weak duality:
$$\lambda \gg 1 \;\longleftrightarrow\; L \gg l_s \;\;(\text{classical gravity})$$
When the CFT is strongly coupled ($\lambda \gg 1$), the bulk theory becomes weakly curved classical gravity. This allows us to compute strongly-coupled gauge theory quantities using simple gravitational calculations, with applications to:
- Quark–gluon plasma viscosity: $\eta/s = 1/(4\pi)$
- Condensed matter systems (holographic superconductors)
- Quantum information and entanglement structure
6. Computing Correlation Functions
The Gubser–Klebanov–Polyakov–Witten (GKPW) prescription gives a concrete recipe. A massive scalar field in AdS with boundary condition $\phi \to z^{d-\Delta}\phi_0$has bulk-to-boundary propagator:
$$K(z, x; x') = c_\Delta \left(\frac{z}{z^2 + (x - x')^2}\right)^\Delta$$
Two-point functions of the dual operator follow by evaluating the on-shell action:
$$\langle \mathcal{O}(x)\,\mathcal{O}(x')\rangle = \frac{c_\Delta}{|x - x'|^{2\Delta}}$$
This reproduces the exact conformal two-point function, confirming the dictionary.
Python: Geodesics in AdS Space
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