Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 1 / b / ii

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- Solution ii

Solution

For a boundary-independent source, the AdS scalar-field boundary asymptotics reduce to

ϕ (z) = J + z^{5} A + o (z^{5}) .

One may therefore extract the source and response directly:

J = z \to 0 lim ϕ (z), A = z \to 0 lim z^{- 5} (ϕ (z) - J) = \frac{1}{5} z \to 0 lim z^{- 4} \partial_{z} ϕ .

The operator expectation value is proportional to

A

; choosing the boundary-operator normalization

O = A

gives the requested expressions. A conventional action normalization can instead multiply this relation by the fixed factor

2Δ - d = 5

.

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