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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-354/1/b/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 1 / b / ii / Solution by

Codex  0 2026-09-29

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.

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