Four distinct points in a conformal field theory with

d > 2

possess two independent conformal cross-ratios, conventionally

u = \frac{x _{12}^{2} x _{34}^{2}}{x _{13}^{2} x _{24}^{2}}, v = \frac{x _{14}^{2} x _{23}^{2}}{x _{13}^{2} x _{24}^{2}} .

(1)

Translations, rotations, dilations, and special conformal transformations can map three points to canonical positions

0

, a point on one chosen axis, and the point at infinity in a conformal completion. The fourth point retains one radial coordinate and one angle relative to that axis. Those two quantities are equivalent to

u

and

v

; the stabilizer rotations remove all its remaining angular coordinates.

For identical scalar primary operators of dimension

Δ

, conformal covariance fixes a kinematic prefactor and leaves an arbitrary function

G (u, v)

F (x_{1}, x_{2}, x_{3}, x_{4}) = \frac{G ( u , v )}{( x _{12}^{2} x _{34}^{2} ) ^{Δ}}

(2)

for one common convention. Since a generic conformal field theory has unrestricted dynamical dependence on both cross-ratios, no lower-dimensional set determines the function. Therefore

dim S = 2 .

(3)

b

 0  0

i

 0  0

Solution

 0  0

For a scalar operator of dimension

Δ = 2

in a boundary theory of dimension

d = 3

, the AdS scalar mass--dimension relation gives

m^{2} = Δ (Δ - d) = - 2

(1)

in unit-radius

AdS_{4}

. In Poincare coordinates,

d s^{2} = \frac{1}{z ^{2}} (η_{μν} d x^{μ} d x^{ν} + d z^{2}),

(2)

so the free Klein-Gordon equation is

[z^{2} (\partial_{z}^{2} + □_{2 + 1}) - 2 z \partial_{z} + 2] ϕ = 0 .

(3)

This mass is exactly that of a conformally coupled scalar field in four-dimensional anti-de Sitter spacetime. The Weyl transformation

ϕ (x, z) = z ψ (x, z)

(4)

therefore maps the equation to the flat half-space wave equation

(\partial_{z}^{2} + □_{2 + 1}) ψ = 0, z > 0 .

(5)

ii

 0  0

Solution

 0  0

The AdS scalar-field boundary asymptotics are

ϕ (x, z) = z^{3 - Δ} J (x) + z^{Δ} A (x) + \dots = z J (x) + z^{2} A (x) + \dots .

(1)

Thus the holographic dictionary becomes especially simple for the flat field

ψ = ϕ / z

J (x) = z \to 0 lim \frac{ϕ ( x , z )}{z} = ψ (x, 0),

(2)

⟨ O (x)⟩ = (2Δ - d) A (x) = A (x) = z \to 0 lim \partial_{z} (\frac{ϕ ( x , z )}{z})

(3)

up to the overall normalization chosen for the bulk action and possible local counterterms. The leading mode is the source and the normalizable subleading mode is the response.