UPI

Holography

AdS/CFT

Gravity in an asymptotically anti-de Sitter bulk is dual to a conformal field theory on the boundary. Maldacena (1997) is still a conjecture — status HYP — and it is the most tightly tested duality in string theory. The geodesic on this page is the Ryu–Takayanagi formula in AdS3: bulk length equals boundary entanglement entropy.

AdS3 / CFT2 · verification_type: software_test

Ryu–Takayanagi geodesic

DER

Drag the handles on the conformal boundary. The bulk geodesic γ_A is the unique curve homologous to region A. Its hyperbolic length is the entanglement entropy of A — that is the dictionary, in AdS3.

Opening φ
1.350 rad
Hyperbolic length of γ_A
5.4212
S_RT = Length / 4G_N
5.4212
S_CFT ≈ (c/3) ln((2/ε) sin(φ/2))
5.4974
Brown–Henneaux c = 3L/(2G_N)
6.00
Cutoff ρ
0.92

Pedagogical units L = 1, 4G_N = 1. S_RT and S_CFT share the same divergence structure; they need not print as equal at a finite cutoff. This is AdS3, not our sky. Open the RT record.

  1. ESTestablished
    Anti-de Sitter spacetime

    Maximally symmetric Lorentzian geometry of constant negative curvature: Einstein's equation with Λ < 0. It has a conformal boundary. This is a solution, not a claim that we live in AdS.

  2. ESTestablished
    Conformal field theory

    A quantum field theory invariant under conformal transformations (and, in the holographic setting, typically a large-N gauge theory). In two dimensions the symmetry algebra is Virasoro.

  3. HYPtheories
    AdS/CFT correspondence

    Maldacena (1997): a gravitational theory in asymptotically AdS_{d+1} is dual to a d-dimensional CFT on the conformal boundary. The original example is Type IIB string theory on AdS5 × S5 dual to 4d N=4 Super Yang–Mills. Unproven as a theorem; tested in a large web of matched quantities. Status HYP.

  4. DERtheories
    Ryu–Takayanagi entanglement entropy

    In AdS/CFT, the entanglement entropy of a boundary region A equals the area of a bulk extremal surface γ_A homologous to A: S_A = Area(γ_A)/4G_N. This is the holographic entropy formula — area as entanglement, not the Schwarzschild entropy of a laboratory quantum.

  5. STOPtheories
    Our universe is not asymptotically AdS

    Observed cosmology is accelerating, consistent with Λ > 0 (de Sitter-like). AdS/CFT requires a negative cosmological constant in the bulk. Applying the Maldacena dictionary to the sky STOPS until a controlled dS/CFT or other dictionary exists.

The dictionary

Bulk
Einstein gravity (plus strings / M-theory) in AdS
Boundary
CFT generating functional
GKP–Witten
Z_bulk[φ₀] = ⟨exp ∫ φ₀ O⟩_CFT
Ryu–Takayanagi
S_A = Area(γ_A) / 4G_N
AdS5 × S5
Type IIB ↔ 4d N=4 SYM
AdS4 × S7
M-theory on the 11D Freund–Rubin vacuum ↔ ABJM

This is the in-domain holographic entropy — entanglement of a boundary region, dual to a bulk area. It is not S_BH of a laboratory frequency quantum, and it is not a photograph of de Sitter sky.

Open the measure labOpen the correspondence