Translation and rotation invariance make the two-point function depend only on

∣ x_{12} ∣

. Scale covariance then fixes its power, and inversion or special-conformal invariance requires equal dimensions for a nonzero scalar two-point function. For identical scalar primary operators,

⟨ O (x_{1}) O (x_{2})⟩ = \frac{C _{2}}{∣ x _{12} ∣ ^{2Δ}} .

(1)

Only the normalization

C_{2}

depends on the operator convention.

Solved by gpt-5.6-sol high.

Solution

 0  0

The same symmetries fix the scalar three-point function to

⟨ O_{1} (x_{1}) O_{2} (x_{2}) O_{3} (x_{3})⟩ = \frac{C _{123}}{∣ x _{12} ∣ ^{Δ_{1} + Δ_{2} - Δ_{3}} ∣ x _{23} ∣ ^{Δ_{2} + Δ_{3} - Δ_{1}} ∣ x _{31} ∣ ^{Δ_{3} + Δ_{1} - Δ_{2}}} .

(1)

Each insertion acquires total scaling weight

2 Δ_{i}

from the two distances containing it, while special conformal transformations leave no independent cross-ratio for three points.

Solved by gpt-5.6-sol high.

Under the AdS-CFT correspondence, each single-trace operator is represented by one bulk field. The leading two-point Witten diagram is a bulk propagator joining the two boundary insertions, equivalently two bulk-to-boundary legs contracted through the quadratic bulk action. The leading connected three-point diagram has one bulk cubic vertex integrated over AdS and three bulk-to-boundary propagators ending at